answering as prompted in your other question...
To be honest, assuming your starting world is anything like our world, getting anywhere near 50% take-up sounds beyond fiction.
Going through your suggested reasons:
Attractiveness – most of us – even if we aren't particularly attractive – are happy, and it's likely to look like some furry fetish if we transform into flawless animal-people. Also, cosmetic surgery already exists, but most people don't take it up not due to price, but because they don't want it.
Healthy body – some might take it for this reason, but a quick google suggests only a very small proportion of the (non elderly) population has a life-limiting physical disorder.
Forget trans; even optimistic estimates suggest only 1% has "some degree" of non-binary conformity. It's not going to give you a significant proportion. Additionally, I'd imagine someone who identifies as a female human isn't going to want to be a female cat any more than a male human
Prefer the species – I think many of us might wonder what it's like, but not want to do it long-term. The grass is always greener on the other side. I imagine a lot of people might take the 1 week trial each time, then revert.
Initial Offerings
For the first offering, anyone even vaguely sensible would avoid an offer of a species change. There's too many risks & unknowns. Will it work? Are there side-effects? Are they really killing people and replacing them with beastman puppets? What happens to my memories & personality?
There's enough people who avoid the first release of an OS because it'll be problematic, how much more an unproven supernatural species change! A 1 week trial is also worth nothing on the first offering, as there's no proof yet that you can change me back!
So once you've had at least a couple of offerings, and people are starting to believe it's for real, and doesn't have any major immediate side-effects...
It's likely that the first people who take it up will be people with a furry fetish, or sex-workers who want to / are forced to monetise that. So that immediately means that – unless furry fetish has suddenly become respectable & desired (it won't in that time – look how long it's taken LGBT to become acceptable) – most people are going to look at it in a somewhat dim view.
I think this is worth considering, because those who first embrace it are likely to set the tone for the overall thing.
Breaking it down
Let's deal with some easy demographic blocks...
I'm going to skip 'Christian' (31%) because it's a very varied block.
Wikipedia gives ~24% of the world's population as Islamic, and most of that will be in countries with some degree of Islamic law. Taking Indonesia as an example (it's one of the most liberal Islamic majority nations, and one I'm somewhat familiar with), the culture is still largely conservative, and species shifting is very likely to be seen as Haram. This is doubly so if species shifting is seen as a 'western' thing – which is likely will as the countries most likely to give it good press are western – or a fetish. So religious observance and social pressure will prevent them from taking up the offer – and in many Islamic nations, there would be punishments (either official or done by local communities) for anyone who did accept the offer.
China (18%) will be your next largest block. It seems unlikely that the Party would look favourably on species shifting. It would likely be seen to go against the One China value. The current party is more traditionally communist, and has been stamping out anything seen as foreign / disruptive (e.g. clampdown on Christianity in China), so I think we can safely assume they would similarly take a very low view of species-shifting. It would also possibly be seen as disrespectful to one's family / community. It would be a career-limiting move.
Your next biggest demographic block is Hindus (15%). My knowledge here is rather limited, so perhaps a commenter who knows more might help. For lack of anywhere better to start, comparing attitudes to LGBT issues, Hinduism is more accepting than Islam, but it's still not as accepted as in the west. Hinduism does of course have a big connection to anthropomorphic animals in its gods – I suspect that would make it reject shape-shifting, as e.g. it would be seen as disrespectful to Ganesha to accept an elephant trunk. On the other hand, it might mean a larger uptake. However, again, very strong family pressures are likely to prevent people taking up the offer.
We've now dealt with 57% of the world's population... You'd now need near 100% uptake in the rest of the world to reach your goal.
Let's go to the other extreme – Western Europe and North America (~14%)
Views here are generally very liberal.
I think we can assume that uptake would be largest amongst the young (mid-teens to mid 20s). Younger children hopefully aren't permitted to make life-altering decisions like species-shifting for themselves? Also once you move up the age brackets, views tend to be more conservative. So let's assume it's highest amongst 15-35s (~25% of UK population, probably similar across WE/NA).
Initially, it's going to be a career-limiting move. Even if governments move to make discrimination based on species illegal, discrimination based on sex has been for decades, but most companies still have a huge gender pay gap. It's going to be hard to be taken seriously in a executive meeting with rabbit ears. Based on other discrimination issues, it's likely to take decades for it to become widely accepted.
It'll also be problematic practically – in particular, any changes like tails or wings, or changes to basic skeletal structure (animal legs) will be hugely problmatic. As noted in your other thread, it'll cause issues for clothing. But also chairs. And cars – I won't want a tail if it stops me driving (or makes it difficult / uncomfortable). Modifications to cars – because of safety certifications – will take years. New furniture will be expensive. Similarly medicine – these new bodies might start healthy, but presumably they'll decay like ours. Medicine will take ages to catch up. Will I have to visit a vet to get my tail / wings looked at? Why is my fur/feathers dropping? I've got lice. Vets aren't licensed to work on humans in most jurisdictions.
There's many jobs where it'll be impractical – a safety hat won't fit big ears; a tail will be a liability in a kitchen. Whilst some might give benefits (wings for couriers...?) the majority of tools and workplaces aren't ready to take advantage of it. Paws won't work a mouse and keyboard. And flight is likely to get legislation pretty quickly (think drone laws), due to risks (dropping your bag at 100ft and killing someone), and privacy fears. There might be compulsory registration – knife laws are pretty pointless if gangs have retractable claws. Every nation will consider how it can be militarised.
With the increase of 'far-right' groups, one can quickly imagine "Keep Europe for Humans" marches (EDL, etc.). Short of actual lynchings, this will create social pressures against species-shifting.
Let's optimistically say uptake in WE/NA, amongst 15-35s (where it'd be highest) is 10%. Let's say 5% across WE/NA. That gives you probably less than 1% worldwide uptake. More realistically, it's likely to be an order of magnitude lower for many decades, without some other pressure.
So... what could you do to encourage people to change?
To make people want to change in large numbers, you'll probably have to invent some big issue, like a disease epidemic which the new species (and perhaps some humans if you want to keep them) are immune to. Or large cash payments, or real-estate in the beastman world across the portal, etc. Or a food shortage – perhaps the new bodies are able to digest grass & leaves etc.?
Sorry... that's all really negative! You might want to just skip your story forward a few decades to the point where uptake is now ~10% and starting to spike up? Or just set it somewhere like California or Vegas where uptake might be highest? Or start with a future world with widespread cybernetics & surgical bodymods which would be much more accepting of this offer.
Overtime as more people start to accept the idea that its okay to change races more people will start to change. You can liken it to gay rights, where people are now more open to coming out, where before people would hide it or be very concerned about revealing their sexuality. A mindset change will allow you to let your people changes races without need a solid reason, and allows you to explore conflicts that led up to this stage. – Shadowzee – 2018-04-24T01:01:19.420
1You can also allow people to keep changing bodies and maybe eventually change back to their original bodies. Like once they have changed and a better races more suitable comes along, they can still change. By knowing that the change isn't permanent, many more people would be willing to take the risk, since you won't be missing out on anything better and could recover your original body. – Shadowzee – 2018-04-24T01:03:57.757
2Might be countered by the presence of superpowers, but the first question I'd ask myself when trying to figure out "why would X do Y?" is "how could Y help X do illegal stuff?". You mentioned elves. I think being able to move/climb swiftly, high endurance and other elf tropes would fit a thief, for instance. – Jenayah – 2018-04-24T01:03:58.690
1@Jenayah There’s actually an even bigger way these bodies affect crime. Since the body’s genetically randomized and for that one week you can shift between them at will, anyone who wanted to do anything illegal would have total anonymity if they treated one of their bodies (obviously more likely the new one) as a disposable anonymous alter ego if they wanted to commit a crime or just do something crazy they’d normally be too shy to do. – Jason Clyde – 2018-04-24T01:23:44.357
erm... live stream my awesome abilities/powers on popular social media...please like and subscribe! become a "patreon" to witness the exclusive my grand escape from a cage on CCTV in real time! ;D – user6760 – 2018-04-24T01:38:38.997
#2 is a pretty good reason. Plenty of older people are chronically ill, and seek escape in drugs or online worlds. A chance of avatar-like transformation is much better than several more years of growing agony. – Bald Bear – 2018-04-24T03:29:32.507
1@JasonClyde good point, hadn't thought of that. As o.m. stated in his answer, I hope you have a way to track who's who, otherwise you're in for a lot of trouble. – Jenayah – 2018-04-24T05:37:34.133
"I'm pretty sure it takes a lot to make somebody throw away their entire physical form in exchange for a new one even if it isn't creepy or gross" One word "furries", you'll have plenty of takers, maybe not a significant chunk of the population but plenty to be visible (in society) without having to add anything else to the mix. – Pelinore – 2018-04-24T07:54:38.160
You should dive a little bit into the Gothic subculture to see a lot of people who would like to change their body - not only into elven-like creatures but also orcs and trolls could be appealing, for the same reason there are leather-jackets with spiky rivets. Avoid statements like ...lost the genetic lottery ... though, if you don´t want to sound like a fascist. A lot of people may not appeal to you but are fine with them selves. Other may be supermodels but are never satisfied with themselfes. I bet who´d op in would not have so much correlation with your beauty-standard as you think. – Daniel – 2018-04-24T11:18:26.460
While I like the concept you've outlined here and think you've framed the question well, I have to point out that this isn't a worldbuilding topic; it's storybuilding. You're looking for character motivations, which is something this site considers off-topic. – Frostfyre – 2018-04-24T12:30:04.523
@Frostfyre I was looking at this more along the lines of a cultural thing. What sorts of people would do this, what would the resulting demographics of these races look like, etc. Things I’d have to keep in mind about the way in which the setting evolves as these developments happen. – Jason Clyde – 2018-04-24T12:37:29.250
Do the changed get to choose from choice 2, choice 3, etc? I mean if I prefer elf to human and swapped species this year, I'd be miffed if next year people could choose to be a dragon, but I had to stay as an elf. – DrBob – 2018-04-24T14:32:20.553
@DrBob: Yes, but only one-way. The way the system's set up now, everyone, even people who have already picked a non-human body, get a choice with the next non-human body too, but there are no second shots at human or any of the other forms they passed on. At least not planned right now. If the first one offered was elves and the second was orcs, if one human stayed human and another picked elf, both the human and the elf will still get the option to become an orc when that comes around. – Jason Clyde – 2018-04-24T14:47:46.337
Might help if you gave people more control over what their new body looks like, notably including the option to retain some of their most recognizable characteristics. Transforming into an elf or a merfolk or a dragon would certainly offer some benefits, but if none of my friends or family recognize me anymore, convincing them of who I really am over and over again would be a massive pain. And they'd probably keep acting weird around me long afterward. And that might well outweigh the benefits. – Someone Else 37 – 2018-04-24T17:30:17.487
If you could let people tweak their forms to their liking (if not outright design their forms form scratch), that'd be an even bigger incentive to opt in than if rolling up a random appearance was the only option. There are lots of people who are unhappy with their bodies for many different reasons, and this would give them a chance to fix that. Also, if the genetic lottery can be lost, some people will lose it each time around- and if this keeps them from becoming the race they truly want, that's less than ideal. – Someone Else 37 – 2018-04-24T17:41:38.323
About your point 2, you have also people who haven't taken care of their body: fat, bad teeth, ingrown nails, pimples, stretch marks... Things that take a lot of effort to improve. Lazy people would love "Oh, instead of a year dieting and exercising, I'll take a new body". – Alberto Yagos – 2018-04-24T22:36:55.073
we already have tattoos, piercings, plastic surgery, hormone therapy and sex change operations, just off the top of my head, I think there's already fair share of interest in wanting to be able to change your body, I'm not sure you would need much of a justification for people taking advantage of an even more substantial change – BKlassen – 2018-04-24T22:45:00.853
Discrimination of the "previous" race, gender, or whatever? – Oleg Lobachev – 2018-04-25T21:13:09.703
On the note of item 4, think about how common members of the furry fandom are. I have a feeling just based on that that 4 will account for a majority share of conversions (at least, at first it will). – Austin Hemmelgarn – 2018-04-25T21:18:13.157
1
If you want to explore how the people feel about their choices after a while (both those who changed and those who didn't) I would highly encourage you to watch this TED talk about choice and happiness: https://www.ted.com/talks/dan_gilbert_asks_why_are_we_happy
– glaux – 2018-04-26T11:09:12.603