MacBook Pro doesn't detect boot drive if turned off for ~5-10 minutes
This sounds like a failing NVRAM (Non-Volatile Random Access Memory) chip - it's not holding the boot values.
When you do an NVRAM reset, it clears the pre-boot environment forcing your Mac to search for bootable volumes which it then writes to NVRAM.  After you shut it off, power is removed from the NVRAM chip and as the voltage dissipates, it loses or corrupts the contents of it's memory.  It could be the chip itself or the associated circuitry that supports it - without putting hands on, it's impossible to know.
You have to take it in for service.  There are board level repair shops that can replace the actual chip(s), or you may need to replace the entire logic board.