Should try and boot the PC without a GPU, and ram, or 1 ram dim at a time, also check your USB port's make sure none of them are shorting out, if any of them are , that will make the PC start and shutdown to protect the cpu. And last but not least try replacing the motherboard battery, this is highly unlikely to be the issue, but they only cost like 5 bucks.
You should disconnect all the cables except the 24pin and the 6pin CPU, take out both ram dimms and start it. if it still doesn't start it narrows it down to either the CPU or the Motherboard, if it does work then it's one of the other components that you disconnected. I'm guessing based on what I saw that it's either ram or CPU. Since the ram is basically brand new I'd bet on cpu.
im no professional, but might be the case's button failing or a bad connection between the motherboard and the case, check if everything is plugged correctly or ((if u can)), try a new case if still doest work it might be a problem with the motherboard since u said it was ancient, or maybe reset bios settings by removing the battery from the mobo and power cycle for a minute, then plug the battery again good luck bro
