I have a CUSL2 and that is the clock chip, yes that chip runs very warm with a 1Ghz P3 cpu. Yours could be having bad cap syndrome or PSU issue if it does not power on every time. Have you checked the PSU outputs ? Any bulging or tan residue on top of any of the caps ?

Ok, but it gets blazingly hot even with no CPU or RAM connected! Do you think that's normal as well?

I have inspected all caps, and they look prestine. I tested ESR, but only in circuit. Nothing seems off, and none of them seem to be shorting either.
I also measured the PSU rails under load, and they are all within tolerance... PSU has been recapped, and has been confirmed to be stable on other mobos.

EDIT: Also worth mentioning that the PSU doesn't turn on either, but I can still measure 3.3v going to the "power on" header. When you short the pins, absolutely nothing happens. The LED on the mobo is still glowing.

Am sure you already checked but what CPU are you using ? If BIOS was not updated to 1007 then it will not turn on if CPU is not a Celeron ( i remember using a Celery 366 to flash my board for the 1Ghz to work).
https://www.asus.com/supportonly/CUSL2/HelpDesk_CPU

The system turns on sometimes (most of the time, lately), so this can't be the issue. I use a 1GHz Pentium III with the latest 1014 Beta BIOS.

I have found out that if the mobo doesn't turn on after shorting the "power on" pins, dropping it from a small height will actually make it turn on.
Sounds like there is a connection problem somewhere. I tried to carefully warp the board a little, push on some chips, try to look for dry solder joints etc, but no luck.
It's kinda strange, because the system appears to work fine when it manages to turn on, no instability issues and so on.