Solved PC sometimes acts up on boot - should I preserve my GPU?

Hi! Lately I've had a problem with my computer, and I was wondering what you would think of it.
My computer is a bit old (especially mobo, cpu and ram; the PSU was given to me by a relative, I think it's been used for five years?), its specs are:

-MOBO: Gigabyte GA-Z77X-D3H
-CPU: intel i5-3570k
-GPU: Radeon RX 580
- 1 TB hard disk WD Blue
- PSU: xfx xps-650w-sew 80 plus bronze
- 8 GB of ram DDR3 (2x 2GB, 1x 4 GB)

Onto the problem: sometimes when I turn on the computer the monitor won't display anything. I say "sometimes", but to be honest recently this has only happened twice (and in general, I only think it has happened one other time half a year ago): once last week, in which case after disconnecting and reconnecting the displayport cable it started working again, and one other time yesterday, whereas it wasn't that simple.
Yesterday I wouldn't get anything on screen, no matter what I did: I tried rebooting it many times, connecting it to another monitor, to hdmi rather than displayport, tried to change ports, still nothing, the monitor wouldn't detect anything. The motherboard would do the usual single beep of "Everything is fine", but I noticed that the fans on the GPU would keep on spinning without stopping, whilst normally they would stop as it gets to windows.
I then tried connecting the hdmi port on the motherboard to use the cpu's graphics. This time it did show something, the initial "Gigabyte" screen, but after the loading on the Gigabyte screen the monitor would go back to "No signal", so I couldn't get to Windows. I removed the GPU from the mobo, tried again but I encountered the same behaviour.
I tried booting from an usb with Linux on it and it worked (still without the GPU)! I could use the computer with the iGPU on linux booted from usb, but for some reason not on windows.
So I backed up everything I needed, and today I tried connecting an old graphic card: this worked and got me into windows! So I took it out, tried connecting once again the original GPU, the RX 580, and it showed only the gigabyte logo, then the monitor went black; I tried powering the computer down (with a short press, not a long one), but it didn't shut down. I waited a bit, pressed it again, did this once more and on this third attempt the computer rebooted itself and now...it's working, got into windows, and I'm writing from this very pc at the moment.

To be honest, I have no idea what's happening: why the iGPU would not get me to windows, if all of this is the graphic card's fault, the mobo's fault, the PSU's fault, if it's just old hardware acting up. I have to say that other than this recently I haven't had any other problem: on Windows everything is great, no blue screen, no random reboots, even if I put a bit of pressure on the computer, for example with gaming, it doesn't stutter, everything seems to be fine and I'm pleased with how it works. I've only encountered this problem on boot

I was wondering what to do, and hoped I could get some advice here. In particular, the most important thing to me would be to preserve my GPU, because if/when I upgrade I'd still like to use it, so I was wondering: do you think it's the GPU fault? And if it's not, could it break because of this? And if that's the case: do you think it'd be smart to remove the GPU for the computer, preserve it for when the time for an upgrade comes and in the meantime use an older GPU I have?

I realize that these are "hard questions" and nobody would know what the absolute best course of action would be, nonetheless I would really like getting someone else's opinion on all this, because to be honest I have no idea where to go from here and any advice or even opinion would be greatly appreciated.

I'm sorry for the long post and for the English, thank you for reading it!

Hello phillpower2,

Thank you for answering! Yes I do remember that from the older thread, and thank you once again for helping me back then. As the other PSU (the corsair CX 500) was acting up when the computer was on Windows by random rebooting, because this one hasn't I didn't think it could be the culprit, but now that you pointed out its age I'm reconsidering things haha.

I would have to buy a new PSU to change it, but you did give me an idea. I think I'm going to do the following: as you suggested, I'm taking out the GPU and then I'm installing an older one (a radeon 7570): worst case scenario the PSU dies but at least it doesn't "fry" the RX580. I'll keep you updated in case anything else happens with the older GPU.
After that, I'll start looking up new PSUs. The idea is: even if I won't upgrade in the nearest future, I can still buy one that I can start using on this older pc and then transfer to the newer pc, right? I hope/don't think it would wear out by doing this. I'll see if I can thing anything suitable.

Does this sound right? Thank you once more!

I've put the GPU out and plan to keep it out until I get a new PSU, so far so good!
About the PSUs: I searched a bit, and found a Bronze 650W EVGA PSU for what I think is a good price (around €57). From what I understand, 650W should be more than enough to use it in a future upgrade of the computer as well, but I have no idea about how "good" the PSU itself is.
It says that the number of the model is ‎100-BR-0650-K2, but even with this information I couldn't find reviews about it (maybe i've been searching wrong?) - do you maybe have any advice on how to understand if such a PSU is good or not? If it can be helpful, this is the amazon.it link where I've found it.
Thanks in advance!

That's a bummer (although it makes sense), I think I'll have to wait for a sale in order to get a gold efficiency one.
Once I'll get one I can put in the RX580 once again and rule out the PSU, right? Or, at the same time, I could confirm it's the PSU if I'll have the same problem while the GPU is still out. So far (since the swap of the GPU) I've had no problems, even though as I stated even with the other GPU it was a rare problem.

I see, thanks. I'm not in a rush to find out about the GPU, and since I can manage to do almost everything I was doing before with the current GPU (which, as old as it is, I'm okay with losing in case the PSU pops) I think I will stick the plan of waiting for a sale to get a gold-rated PSU and then trying again with the RX580.

Since that could take months, I think the post can be closed for the foreseeable future. In case anything problematic happens - I get the same problem with this older GPU, or with the newer PSU whenever that happens - I'll make sure to ask an administrator to reopen the thread so that at very least it can be updated.

Thank you phillpower2 for the help, opinion, and for giving me an idea on how to proceed!