My HP Pavilion will only start after 2-3 reboots. Every time

Question

I have a HP ENVY h8-1514eo stationary PC, model D7A59EA, and it has the weirdest problem. When I boot it, it always fails the first 1-3 times. This makes it impossible to run Windows updates since that requires a reboot and it always fails on that reboot. I have to manually turn it off and on a couple of times in order to get it up and running again. I have asked this question on HP's support forum but I haven't had a single answer in 3 months and generally SuperUser is a lot faster than that :)

Now, what I have noticed is that every time it fails I get the same loop in the disc-icon led. It blinks very fast about 8-10 times, goes dark for half a second, and then blink once more. Then does it over and over and over again until I forcefully reboot. Here is a video showing it:

https://www.youtube.com/watch?v=j1V1jZWKpUQ

This was a problem that started to occur after I purchased a new monitor (Acer 27" Predator XB270HU G-Sync), but I can't see how the monitor would ruin the computer. I did notice that the computer tends to start when I unhook the monitor prior to start, but that also feels a bit superstitious.