now you have ruled out the plug and can stop looking at the fuse, the timings with the fuse and unplugging it where just coincidental.
now its sound like a dry joint hopefully on the PSU psb, now to check this.
you need a multimeter set to DC voltage.
check all the pins on the output on the psu after you unplug it for 5 minutes so its working again.
Power Supply @N64
Pin Name Direction Description
1 GRND Ground
2 Vcc 3.3V 2.7A
3 Vcc 3.3V 2.7A
4 GRND Ground
5 GRND Ground
6 Vcc 12V 0,8A
check they match.
1:- then use the n64
2:- when the power dies DO NOT UNPLUG IT, (in its case not open as in below photos)
3:- remove PSU from N64 and check voltages as fast as you can, if all voltages are OK reconnect the PSU to N64 and confirm it does NOT work,
4:- If the N64 works again then keep repeating until either one of the voltages is missing/low or the N64 does not work when PSU is working,
this will rule out N64 or PSU issue and point to which is the issue.
Hence, if
1:- all the voltages are correct but N64 does not work then its N64
2:- One of the voltages is not correct after N64 dies then its PSU