If you can get the motherboard to reliably move through (some) POST codes when you press on the CPU vs. the motherboard doing absolutely nothing when you don't press on the CPU, then, you may have cracked joints under the BGA CPU socket. Some stock Intel coolers are notorious for warping the board when they are fully tightened. This can be made worse if the motherboard was shipped with the CPU cooler still strapped to the board or if the previous owner ever allowed the CPU to overheat (due to HS clogged with dust.)
But it could also be bad contacts / corrosion on the CPU pins... though you said you tried different CPUs, so that should have gotten rid of any in the CPU socket.
I also see the original United Chemicon KZG (brown) electrolytic capacitors around the CPU and OST RLX(?) (blue) capacitors are still present on the board, so there's always those to suspect as well. UCC (United Chemicon) KZG series are known to be problematic, especially 6.3V ones with capacity of 2200 and 3300 uF, even if they don't look bulging (though most frequently, they do bulge when they go bad.) OST RLX series are the in a similar boat, especially the smaller ones.
So try doing at least 10 tests where you try to power the board with pressure on the CPU and without to see how the POST codes behave.
It may also be worthwhile to try putting a heatsink on the CPU but *not* strapping it to the board (to prevent warping) to see if that changes anything. And also, make sure the CPU cooler fan is plugged in. Some boards will sound an alarm if they don't detect a plugged in fan.
Lastly, going from the last sentence in the above paragraph, also check for any missing or damaged SMD components near or on the CPU fan header. Also try some different 3-pin fans, just in case. Some socket A motherboards for sure will stay in RESET state / not POST if a CPU fan is not detected. With P4, this wasn't usually the case... but I always keep it in mind when testing motherboards, just in case.