These could easily be two separate issues.
I'm inclined to agree.
My experience with problematic video cards has been predominantly about heat. With a machine starting up from cold - in the thermal sense - I would separate the two.
Then, there is the possibility that they are related, but I'd bet my $1 that they are not.
You see, I'm not. This now sounds more like a post failure than a power issue. At this point LITERALLY anything can cause a computer to not post. A borked GPU can do that, as I have been dealing with borked GPUs and post failures for the past 5 days (Finally fixed!)
But the PC not wanting to post could be PSU, Bad RAM, a bad peripheral on an ATA/SCSI bus, a bad PCI/PCI-E/Expansion card, poorly seated CPU, and whatever else you could think of. Some times these result in a visual or auditory error, but there have been cases with me that a peripheral just decided to not let the PC boot for some reason. Why unplugging it and plugging it back in works? Idk maybe your shaking the gremlins loose. PCs are complicated parts that have required dozens of engineers to even conceive on a base hardware level, and base concepts are still in use fresh from the early 1980s. As a result you can as a computer engineer come up with a logical and likely solution, to find some bugger off part was not in right.
My failproof suggestion is to unplug everything inside you do not need to start the computer. No drives, minimal RAM, only a GPU (If you have an iGPU or an APU (Intergrated Graphics) use that instead of a dedicated card, or if you are using that regularly use a dedicated card if you have one), and take out any other expansion cards. The point is to unplug as many things as you can think of in order to eliminate most of the components as possible issues. Then if you have the parts, start replacing stuff. I'd suggest swapping out the PSU first. All of this can stem from bad/unclean power or voltage spikes/drops starting or stopping. PSUs are like suspension bridges. You build it right, it will last for decades, you do not build it proper, like the majority of those on the market, you end up with a magic smoke machine.
If you understand how CPUs are installed, you can try reseating it (Or swapping it out, but this is an unlikely issue because it wouldn't run at all of the CPU was in wrong)
At this point you can understand one thing. A part is causing the computer to not turn on. This is all you need to know, because you can most likely replace one thing.