Replace all the capacitors group where you found a bulging one, and then the remaining groups if any. Abits from that era are notorious for capacitor issues.

Need hi-res photo of motherboard.
And solder side too.

440BX is a very simple board.
- 3.3V - for the chipset and memory, may be trace from an ATX PSU direct or from some DC-DC module.
- 2.5V - to power the clocking IC. It comes either from LDO or from PWM IC.
- 1.5V Vtt - from LDO, like 1085 or from a linear regulator on transistors
- CPU power, depends on the set VID. In the absence of a processor, all VID = 1 and voltage should not be applied.

Possible jambs when replacing capacitors.
- the polarity of the capacitors is reversed. Usually on the board, the polarity of all capacitors is the same - the minuses are located in one direction.
- Abit not Asus. Minus means minus.
- bridges from the soldering point to adjacent trace. Look into the lens.
- When soldering, some resistors or capacitors were broken, or the trace were interrupted. Look into the lens.
- Exotic. For a crappy capacitor of unknown origin, the minus may be confused when its marking.

Some photo from Internet, for comparison


and
http://computekus.com/images/Abitbe6ii.jpg
Not https URL, because image not inserting in "img" tag

This will sound crazy, but have you tried a faster CPU fan? My Abit BH7 powers off almost immediately with particular fans connected to the CPU fan header. These fans do not return any value for fan speed. Using a fan from a stock intel cooler results in a successful boot.
Booting with the intel fan, changing some BIOS settings and saving allows the board to then boot with the bad fans connected.
These boards possibly refuse to boot if no CPU fan is connected or detected (likely using fan speed monitoring as a basis for presence detection) to prevent overheating. It's weird that my BH7 allows booting with an "undetected" CPU fan after saving BIOS settings though.

dulu wrote on 2023-07-29, 09:55:

Mobo turns off after 3-5 seconds after pressing power button. The symptom is the same regardless of the used psu, ram, gpu (or no ram, gpu, cpu, bios chip).

same with no bios chip means either short or Abit used some external chip with watchdog?

dulu wrote on 2023-07-29, 09:55:

The bios is working, because there are beeps accurate to the problem (ram error, gpu, etc.) just before shutdown.

Beeping means CPU is running means clock generator and 3.3/2.5/1.5 rails do work
So not a terminal short, more fingers pointing at some sensing/protection chip. sadly I cant find anything about Winbond WB3977EF to know if it has those capabilities. I like the idea its shutting down due to no tacho signal from cpu fan.

dulu wrote on 2023-07-29, 09:55:

Also, just before shutting down, a green light comes on on the monitor, indicating that the graphics is trying to initialize the image.

not initialize, all is needed for monitor to think it should turn on is pullups powering pins 12/15 (I2C eprom)

dulu wrote on 2023-07-29, 09:55:

I started the mobo with bulged capacitors and the mobo started fine until this glitch appeared. Replacing the bulged capacitors did not help.

you need to replace all electrolytic caps, not just the bulged ones

rasz_pl wrote on 2023-08-01, 06:14:

Beeping means CPU is running means clock generator and 3.3/2.5/1.5 rails do work
So not a terminal short, more fingers pointing at some sensing/protection chip.

RC5058 & CS51313 have same function - PWM controller for P2/3 class CPU.
From this we can conclude that one of them is responsible for 3.3V (I think this is СS51313), which is programmed for 3.325-3.425-3.525V.
This chip has pin 13 - Overvoltage protection pin. It can also be processed by the BIOS.

In general, voltages should be measured, not guessed. 😀

i found cs51313 - 3,3V on Vout pin when mobo is running (inaccurate measurement, cheap meter)
but where is rc5058?
do i need to desolder pin13 on cs51313?

dulu wrote on 2023-08-05, 13:34:

do i need to desolder pin13 on cs51313?

Overvoltage protection pin. Goes high when overvoltage condition is detected
on VFB.
Nope.
If pin 13 is high (3.3 or 5V), then with 3.3V all right

I`ld be see on pin 17 RC5058 - PowerGood. Its should be "low"

pin 17 on RC5058 = 5,2V (same as my psu +5V line output)

dulu wrote on 2023-08-05, 15:20:

pin 17 on RC5058 = 5,2V (same as my psu +5V line output)

Sorry, lost in translation 🙁

It is worth measuring the voltages Vcpu, Vtt, V3.3V, V2.5V on the motherboard itself.
But I won’t tell you the measurement points, I don’t have Abit.

It may be worth looking in the direction of overheating - an incorrectly installed radiator.
Or as mentioned above - the fan speed.

so i need info where i should measure voltages 😐

dulu wrote on 2023-08-06, 15:10:

so i need info where i should measure voltages 😐

I have CUBX-L, but I will try to express my assumptions.
1. Dont know (or Vtt)
2. Vtt = 1.5V (or dont know)
3. Vcc - depend of CPU, ~1.5V for tualatin, ~1.7V for cu-mine, ~2V for mendocino
4. +5V from PSU
5. +3.3V from CS51313 and Q8&9
6. +5V from PSU
7. +2.5V for Clock generator IC&CPU.
8. +5V standby from PSU

1 - 1,56V
2 - 2,59V
3 - 2,08V - but cpu is cu-mine 😜
4 - 5,07V
5 - 3,38V
6 - 5,08V (1,5V when off)
7 - 5,34V (2,27V on second/opposite pin)
8 - 5,12V

In appearance, everything is fine.
Except V_cpu.
According to the datasheet on RC5058, 2V is all VID = 1.
Those the CPU is not inserted.
Or the signals from the slotket to the RC5058 do not pass.
Check the jumpers at slotket.

This is a pinout for Tualatin, I have not found another in normal quality.
Therefore, AK36 (Vid25mv) for Coppermine read as Vid4.

PS. For Cu-mine and Tualatin (on a non Tualatin ready motherboard) the VID4 always must be grounded

i have mendocino on slotket and cu-mine normal cpu, not slotket. Both V_cpu=2,08V.
VID4 in RC5058 is not grounded - no connection

dulu wrote on 2023-08-20, 11:19:

i have mendocino on slotket and cu-mine normal cpu, not slotket. Both V_cpu=2,08V.
VID4 in RC5058 is not grounded - no connection

I never heard of Cu-mine with 2V V_core - 1.75max
By the picture, VIDs on RC5058 come from DM74LS374.
This is related to Jumper Free Configuration Control, including V_cpu.
Where do they come from - without a motherboard in front of my eyes, I can not determine.