Try pulling everything off the mother board except the power supply and the jumper for the power switch. This includes memory and cpu.
Some computers do not have an on-board speaker. If this is the case with yours, leave the speaker in the case plugged in.
In some cases I’ve seen small screws drop behind a motherboard and short it out. If you can, pull the board from the case. You can place it on a piece of cardboard while you test to protect the contacts on the back side.
Power up.
If the board (or bios) are ok, you should get some sort of error beep. If so, this is a good sign. Start putting devices in one at a time and power up. As long as you get error beeps, you’re in good shape. In order put in the cpu, memory, video. Once the computer fails to boot, you’ve found your problem.
Indeed, I worked on computers at the local sheriff’s department about 10 years ago and they were filled with this fuzzy brown dusty goop from all the smoking there, not good at all. Probably 1/3 of the computers weren’t working.
Some motherboards wont even boot to BIOS if it thinks that the computer chip fan is not plugged in and working… thats intended to protect a multihundred dollar PC chip from overheating.
(Same if it thinks that the power supply is bad and/or delivering incorrect voltage, but you stated that you already tried swapping power supplies.)
My home computer won’t power up if the motherboard is too cold! Hehe. I used to switch off the powerstrip to the computer desk after shutting down the computer, so that the printer, monitor, and cable modem are all off, too. But after finding out that the motherboard needs to “warm up”, I just leave that power strip on, now.