I did some more testing. SiS5595 seems to have stand by power (5V) so that is good. I measured it from pin 22 (RTCVDD).

There is 5V on the ATX Power ON/OFF control pin 25 (PS_ON#) before pressing the power button. That should be correct since datasheet says that PS_ON# is internally powered by the RTCVDD and power on event will set it to low and start the PSU. And indeed, it goes to low when I manage to start the board by holding down the corners and pressing power button. But it is very hard to measure that way and the power goes off and on even if I move the board a little bit.

I also tested continuity from PS_ON# to ATX connector PS_ON (green wire) and it seems to be good.

Am I missing something? Only thing I can think of now is to try to resolder the 5595.

Do you have to push down on both corners? If the motherboard mounted on a tray with standoffs when you're pushing it? Have you checked for any shorts to Ground on any of the ATX supplies (PSU not starting, or doing repeated start-stops is often due to a short on the board)? I've had a look at your photos and can't just seen anything that looks like it'd cause a short. Of the corners you're pushing it looks like the one near the ESS chip just has the serial drivers. The other end has the CPU voltage regulator, but a problem there shouldn't directly stop anything from working.

I'm not using standoffs at the moment so when I push down both corners, the mainboard straightens close to what it should be. Sometimes it is not needed to push down both corners but pushing down the two opposite corners is a sure way to get the PSU to start. I discovered that method to start the mainboard when I was testing if some of the components get warm. Who knows, maybe the mainboard would work if I mounted it to a case but I would not trust it to operate reliable.

I did test shorts from the ATX connector and there were none.

There are +5V (non standby) pins on three sides of the SiS5595 and the fourth side has the standby voltage pins. Maybe the SiS5595 loses one of the 5V supply voltages, becomes unstable and starts to power off and on when I stop pressing the corners down and finally shuts down.

As you're looking at, PS_ON# could be a cause. May also be worth looking at the PWRBT# line. I guess that should be pulled high by a resistor somewhere (probably near the header pins), and shorted to ground when the button is pushed. Maybe that's not being pulled high properly, so the 5595 sees random button presses. If you haven't already, might be worth spending some time pushing various components and seeing if you can narrow down where the fault might be.

I think I can be 99% sure now that the problem is some loose 5595 pin. I've saved a few old credit cards and other plastic cards since they are good at spreading thermal paste. I cut one card to match chip size so I could easily press down whole row of pins at a time. And it did the trick. All I have to do is gently press down the card like on the picture and the mainboard can be started by pressing a button.

So I really need to resolder that thing now. First I'll need to find something to practice on. At least I now know on what side the problem is.

Flux. Make sure you have flux or you'll end up with loads of solder bridges. Solder sucker and braid for removing solder bridges. If you can narrow it down to which pin then you can just worry about that pin.

Looking at the datasheet there's nothing directly PSU connected I can see on that side, but there are Ground pins on 172, 188 and Power on 168, 174. Plus a clock on 162. Might be worth giving those pins a gentle sideways nudge with a fine screwdriver to see if there's an obvious dry joint.

[edit: also it's sometimes easier to spot problems in a decent photo than when looking at the board, so a photo of the chip and surrounding areas might be handy]

I'm happy to report that the mainboard is now working! Resoldering the 5595 did fix it.

I had an old modem that I used for practice. Practicing first was a very good thing. Yes, lots of flux is needed and braid is necessary too. But I would also add that only a very little new solder is needed. Use too much and lots of pins are shorted. Another important thing to note is that the tip of the soldering iron must not be pressed with force. It will just bend the pins away from pads. The weight of the tip alone is enough.

But practicing is the most important thing. I have two VLB VGA cards that I think needs the same treatment but I'm going to practice some more with the modem before soldering those.

The 5595 soldering went very well. I tried to take pictures of the results but they were not sharp enough. I soldered three sides just in case. I left the side closest to PCI slots to the way it was since it would have been harder to solder because the PCI slots are too close.