If there is no post, and no physical damage to the board, at min replace the VRM Caps.

Before recapping did some testing.
Testing with two P3 CM 1BGHz SL52R, I would get no beeps or post codes on a post code card.
CPU voltage at Q8 and Q10 Drain will show for less than a second then drop to near zero volts just after power on. (Fluke 289) This is a sign that there is short on the CPU VRM.

After VRM recap board posted.
Q1 (Q9)
Drain: ~5 volts. (input from PSU)
Source: ~1.78Volt (CPU voltage)
Gate: ~ 5.3Volt

Q8 (Q10)
Drain: ~1.78Volts. CPU voltage. With the SL52R, Should be around ~1.75Volt
Source: 0.02v
Gate: ~6.77volts

Tested all the old caps with a Keysight U1733C LCR meter. Every single VRM cap was shorted. Both on the 5volt supply and CPU output side.
As others have posted it is strongly recommended to recap the whole board.