GBA can power-on once, and won't turn on again for a period of time

I purchased a lot of Gameboy Advance systems to refurbish, most of them couldn't power on due to degradation of the power switches. I bought some replacement power switches, and one system in particular is giving me trouble.

I replaced the switch on it, and it came to life, however I quickly found out that it stopped working after I had powered it off. I tried to power it on again, and only the power light flashed, one more time and nothing happened at all.

Thought it might have been an issue with the new switch I installed on it, so I replaced it, but I still have the same issue. I did swap through some various batteries, still no difference.

It has been sitting on my desk for awhile now, and I noticed that if I let it sit for a short amount of time, it will consistently be able to power on once again but only once, then I only get the power light to flash before the system is completely dead again.

What could this be? This is the only system out of the lot having this issue and im stumped.