I am back, and I finally took some time to clean up my computer.
Turns out, the NVMe SSD that was causing the problems already had an heatsink that came with the mobo, but when I built this computer, almost 5 years ago, I forgot to remove the sticker from the heatsink 
Now I removed it, and just to be safe I also unplugged and replugged the SSD, just in case it was a loose contact or something.
Also, one of the case fans wasn’t spinning, but that only blew air on the PSU, so while definitely not a good thing to have it stopped I don’t think it was relevant to the problem. After some cleaning it started spinning again, but if it stops working again I’ll just replace it.
Now I’ll just have to wait and, if the problem doesn’t come up again, I’ll mark this thread as solved. Thanks everybody for now.
OT: Not gonna lie, but now I want to know why M43K8M’s message was hidden. Your discussion on hibernation and swap space was interesting!
And again after I cleaned up my computer it didn’t happen for long enough to give me hope…