So, this socket 7 motherboard has been driving me insane for months.
This was the behavior when I got the board:

The keyboard was connected, but keyboard LEDs didn't flash and there was no response.
The CPU being used is an IDT WinChip C6 200, so it is misdetected. The reported amount of RAM is also incorrect, I'm using a 16+16+8+8 combination.

After flashing the newest (HDD patched) BIOS in a different board, this is the result:

When it hangs, it is always at the same number. The CPU is now properly detected.
I have successfully entered the BIOS once and I was able to move around with the keyboard for a few seconds before it locked up so the keyboard controller is not completely dead.
I can't get to the BIOS anymore. The board also doesn't boot up reliably, only giving a video signal about 50% of the time. The keyboard fuse is fine.

Using only one bank (either one of them) and trying different RAM doesn't fix it.
RTC is not the issue because I have modded it to use a CR2032.
Proper CPU voltage has been selected with jumpers and measured with a multimeter to be 3.50 V.
There is 256K of onboard cache and a socketed chip with an additional 256K. Jumpering the board to use only 256K/removing the extra chip does nothing. The cache doesn't get hot during operation.
I have tried three different PSUs - a 500 W FSP, 400 W no-name and a 420 W Spire, no difference.
I have checked virtually all ceramic caps for a short to ground and found nothing.
The electrolytes don't seem to be a problem either, I desoldered a 220 uF one (CPU Vcore) and it measured to be 205 uF, 0.34 ohms ESR.
I even soldered a 3300 uF cap near the RAM slots (3.3 V line), no difference.

Any final ideas before it goes to the junk pile?