Have the BIOS ROMs been checked?
Long term lack of powering up EEPROMs can cause the contents to bit rot, several video cards of that era need reflashing to work again.

Hard to say if it is just the caps.
I would do further diagnosis with something like a POST code card first before jumping to conclusions.

Excelsior wrote on 2023-07-03, 09:13:

ASUS P4P800

I hope for you it's not the south bridge (flawed USB ESD protection design).

Excelsior wrote on 2023-07-03, 09:13:

I recently received 2 motherboards (ASUS P4P800 and Gigabyte 7VT600).

Hi,
Check temperature (I'm using just my finger) of south bridges (and maybe other chips) - they shouldn't be scorching hot.
If you upload hi-res photos maybe someone will spot something wrong.

+1 for reflashing BIOS EEPROMS.

Excelsior wrote on 2023-07-03, 09:13:

Is it possible that one or several capacitors might be actually dead even if they look normal at exterior? Is there a way to check each capacitor?

It is entirely possible for capacitors to look intact but be defective. They can only be reliably checked out of circuit unfortunately.😐

Roman555 wrote on 2023-07-03, 11:32:

Hi, Check temperature (I'm using just my finger) of south bridges (and maybe other chips) - they shouldn't be scorching hot. If […]
Show full quote

Excelsior wrote on 2023-07-03, 09:13:

I recently received 2 motherboards (ASUS P4P800 and Gigabyte 7VT600).

Hi,
Check temperature (I'm using just my finger) of south bridges (and maybe other chips) - they shouldn't be scorching hot.
If you upload hi-res photos maybe someone will spot something wrong.

The southbridges are a little warm but that's all, not hot at all.
https://imgur.com/a/8PJLbZ0

Excelsior wrote on 2023-07-03, 13:17:

Roman555 wrote on 2023-07-03, 11:32:

Hi, Check temperature (I'm using just my finger) of south bridges (and maybe other chips) - they shouldn't be scorching hot. If […]
Show full quote

Excelsior wrote on 2023-07-03, 09:13:

I recently received 2 motherboards (ASUS P4P800 and Gigabyte 7VT600).

Hi,
Check temperature (I'm using just my finger) of south bridges (and maybe other chips) - they shouldn't be scorching hot.
If you upload hi-res photos maybe someone will spot something wrong.

The southbridges are a little warm but that's all, not hot at all.
https://imgur.com/a/8PJLbZ0

It looks pretty good. Try to run the mainboard without RAM. It should beep. A PCI debug card is usually helpful in such cases.
Also check voltage of a CMOS battery and replace if it's lower than 3v. I doubt it will help but it's worth to try.
If you have a multimeter you can make some measurement: CPU core voltage, bridges voltage

Schematic manual GA-7VT600 (a different rev. but it's better than nothing)

Roman555 wrote on 2023-07-03, 16:24:

It looks pretty good. Try to run the mainboard without RAM. It should beep. A PCI debug card is usually helpful in such cases. A […]
Show full quote

Excelsior wrote on 2023-07-03, 13:17:

Roman555 wrote on 2023-07-03, 11:32:

Hi,
Check temperature (I'm using just my finger) of south bridges (and maybe other chips) - they shouldn't be scorching hot.
If you upload hi-res photos maybe someone will spot something wrong.

The southbridges are a little warm but that's all, not hot at all.
https://imgur.com/a/8PJLbZ0

It looks pretty good. Try to run the mainboard without RAM. It should beep. A PCI debug card is usually helpful in such cases.
Also check voltage of a CMOS battery and replace if it's lower than 3v. I doubt it will help but it's worth to try.
If you have a multimeter you can make some measurement: CPU core voltage, bridges voltage

Schematic manual GA-7VT600 (a different rev. but it's better than nothing)

I tried without RAM, without video card, only with the processor but there was no beep.
However every time the system starts there is an almost inaudible sound like a click but it is very faint, you have to be close to computer to hear it.
There was only a little dust on the MOSFET but no damage, I cleaned with a piece of paper and it is now like new. The battery is new and I also tried with other batteries but there was no change.
Thank you for the suggestions, I will check the voltages.

Regarding 7VT600.
Check the die of the CPU, it is quite fragile.

It could be damaged even during transportation with a slight distortion of the cooler position.

This led doesn't look happy:

Excelsior wrote on 2023-07-03, 19:17:

However every time the system starts there is an almost inaudible sound like a click but it is very faint, you have to be close to computer to hear it.

that click is often a short on the motherboard, if power supply keeps shuts down its one of main rails, if it keeps going its cpu/ram power converters
both boards are from the middle of cap plague, might even worked when put away but dried since then

After eliminating all other possibles causes (ICs, traces etc.) it seems that the only remaining causes are either the southbridges or the capacitors. The boards themselves look good and no exterior hint is visible on them.
The problem is that all the capacitors look normal, there are no bulged or leaking capacitors.
Is it possible that one or several capacitors are in fact dead/dried even if they look normal at exterior?
If so what is the best solution: replacing all the capacitors on the boards or only replacing the most likely problem making capacitors? Which ones are the most likely to be dead - those near the CPU sockets?
If the southbridges are the problem then there is nothing that can be done. But in this case when the systems are powered the southbridges are only a little warm, not hot and not cold. And they remain a little warm as long as the systems are powered-on, so it seems that the southbridges might not be the problem.

Check the die on the P4P800's northbridge.

I know it sounds weird but I had a Gigabyte 8IPE1000-G not POST, SB was normally warm (as in, not too hot to the touch) but no POST or any activity. Come to remove the NB heatsink and notice the chipset die had a microscopic hole in it. In hindsight, it should've dawned on me NB was bad as it was getting unusually hot.

"looking" at electronics doesnt eliminate anything. You mentioned checking voltages, but didnt follow up. You say everything looks great yet Roman555 and Karbist found cracked components