nvme storage should always have a heatsink or the storage will die due to fast speed of the drives they cause alot of heat.
I have several M.2 NVMe drives running at Gen.3 speeds quite happly for several years with no heatsinks. I prefer to use a heatsink on Gen.4 drives but I'm not paranoid. I don't have a Gen.5 in my latest build, but would definitely fit a heatsink for this type. Some drives run hotter than others. Some run cool.
If you push an M.2 drive really hard with multiple read/writes and no heatsink in an enclosure with poor ventilation, it could reach the "throttle" point somewhere over 70°C (typically above 80°C). At this point the M.2 drive's controller chip will slow down data transfers, in an attempt to reduce temperatures.
Looking at my "Samsung 980 Pro with heatsink" in Hard Disk Sentinel, I can see it reached 74°C at some point in its life, probably during a 4K video render. I use this drive as a scratch disk and its current temperature is 36°C. The drive is installed in the third M2 slot which is not covered by the motherboard's heatsink.
My other Samsung 980 Pro cooled by the motherboard heatsink has only reached a maximum of 56°C and is now 30°C. I guess that means the big heatsink on the motherboard with its own tiny cooling fan is doing a better job of cooling the 980 Pro that came without a heatsink than the 980 Pro with integral heatsink.
My Windows boot drive (Kingston SKC3000) also on the mobo heatsink has peaked at 55°C and is now 29°C.
Will I have any issues transferring the CPU to a new motherboard due to what happened when I built the PC back then? Should I be worried it might break?
Just be careful when installing the old CPU into a new mobo. Place the CPU carefully on top of the socket and "wiggle" it gently until it drops in place. Do not press down hard on top of the CPU. You could bend the pins again and they might snap off. The more times you straighten bent pins, the more likely they are to suffer metal fatigue. If you are sufficiently skilled, you can solder new pins (from a donor CPU) back on the damaged CPU. It's tricky work, because adjacent pins can become unsoldered!
I hope your 3600's pin(s) weren't this badly bent before you fixed them. I've had similar when buying second-hand CPUs on eBay.
Could this possibly be caused because of something faulty in my mother, like for example the broken pin?
If a pin in your 19-way USB3 header is bent and touching a nearby pin, you might have a short cicuit (or partial short circuit) from +5V down to 0V (ground). If so, that might explain why one PSU works and another one doesn't. The PSU which trips might have a more sensitive over-current detector on the +5V rail. Make sure there are no pins touching in the 19-way header.
https://www.reddit.com/r/pcmasterrace/comments/kzyt1f/i_bent_a_pin_on_my_usb_30_header_and_broke_it/