As several have noted, this is in fact the normal state of affairs for some generes. I think the association of pop music with a specific performance is a rather recent phenomenon. How might that not have happened?
First, look at things like “show tunes”. The emphasis is on live performances and it’s a large scale collaboration. The lead singer and conductor change independently several timee during the original run, and then multiple troupes perform it on different tours.
Eventually people do release recordings, but the song is already famous before the specific recording makes the rounds. After a while they become “American Standards” associated with a composer, not a specific artist.
Maybe the culture is such that most interesting music comes from this, and songs produced in isolation are generally deemed empty of context.
Second, the music publishing industry might have went down a different road. They could release the same composition with different performances as a matter of course. There could be different market segments distinguished by geographical region, ethnic groups, age, or other factors. Make it other than geographical so the segments are mixed. Then, it makes sence that they make a point of releasing the same song to different segments rather than totally different pieces, so it serves as a unifying influence and drives cross-segment sales.
Some of the market segments may still have a culture of live music and local favorite performers. The publisher sells the sheet music and other tie-ins, not just a recording. People hear the recording on the radio and also the same popular songs on the street and in local gatherings.
Finally, composers may intend for different performers to put their own interpretation on the song. When a song is popular, different performers will compete with the same song.
In short, ensure that popular music (whatever it sounds like) still has essential characteristics that orchestra, opera, and musical-play has.
Most DJ's pretty much fill that role. Eg David Guetta. He writes the music, programs it, and gets an artist to do the vocals. But people buy "his" albums, even though most of the actual musical work is done by others. Pitbull is a lesser example, as most of his music is actually collaborations with other artists, under his name. – Lu22 – 2017-01-03T06:30:47.410
how are you getting past copyright infringement of the lyrics? – Keltari – 2017-01-03T09:55:28.283
That's the case of classical music IRL. Likewise “show tunes” and opera. I have 3 different recordings of South Pacific, several with overlapping American Standards, multiple interpretations of the Pines of Rome… – JDługosz – 2017-01-03T10:43:13.383
«music and lyrics writers to become at least as rich and famous as performers.» What, Andrew Lloyd Webber isn’t rich enough to qualify? – JDługosz – 2017-01-03T11:10:41.983
What immediately comes to my mind is the song "Hallelujah", written by the recently deceased Leonard Cohen. It's been recorded probably hundreds of times in recent years by a variety of performers, but isn't strongly associated with any particular singer. It's rare for a song to enter the public consciousness that way, but it does happen once in a while. – Charles Burge – 2017-01-03T23:56:24.443
Not a pop music expert, but I think a lot of Bob Dylan songs were (are?) better known for their performances by any number of other people. And then there's "Stairway to Heaven"... – jamesqf – 2017-01-04T03:54:39.847