Bios seems like a premature conclusion.

Got multimeter?
Can you confirm voltages to things?
Eg the transistors.

Can you smell anything anywhere?
If not, give it a wash and take clearer full res pics.

Thanks for the reply, myne!
No, no funny smells. Just smells like a motherboard 😉

I did some more testing with my multimeter, and made some observations. The electronics are getting a little over my head, so I have no idea if any of this means anything.
The resistence between 5v and gnd seems kind of low. It's low enough that my multimeter beeps as if it has continuity. Getting (if I'm reading it right) about 50 ohms.
Some of the transistors are doing stuff, some of them are idle. The big one in the middle is functioning, as well as the TL431 next to it. One out of the three transistors between the isa slots is idle, and all of them up near the jumper block and headers are idle. One of the transistors near the BIOS chip is doing stuff, but the other one is showing low voltage - around .5v. Is this significant?

But maybe (or maybe not) of most interest is around the clock generator chip and the crystal. I found that if I shorted some of the pins to 5v or ground (depending on the pin), I could make the PC analyzer card show a clock signal, and irdy signal, and turn off the reset signal. Still no BIOS codes though...

The mobo is pretty clean. I think the gain on my camera is making it look dirtier than it is. Here are some more photos that are hopefully better:

With 5v resistance that low, I'd expect smoking.
Anything getting hot?
Can you mark the tested points on the picture?

Do you get the same results with everything that can be unplugged, unplugged?

Based on my experience, those cache chips don't look legitimate. I'd say with a 99% certainty that they're fake. I'd remove them and continue the tests without them, as they can introduce further instability.
The good thing about this board is that you get the sockets, so that if you finally get it working, you can later add real cache. Not that it makes a huge difference but it's nice to have it =)

Check the C3420 transistor (the big one). It's usually the culprit. It produces the voltage for the CPU. Carefully (don't short it) check if the voltage is OK.

If there is no clock signal, nothing on the board will work. Do you have an oscilloscope? Confirm that there is indeed no clock signal and begin by repairing that. Trace it from the crystal oscillator, if there is no output from that everything else downstream will not work.
Until you know that you have a good clock signal (in addition to good voltages) you can't meaningfully diagnose anything else.

Myne:
Nothing is hot or smoking. I got my resistance reading right on the power connector and various points around the board. If I marked all the spots I tested on the picture, it would be more marks than picture!
I tested all the 5v pins with all the ground pins on the power connector, and I tested ground from multiple spots on the board with every spot on the board I knew or suspected should be connected to 5v. Everywhere was showing this low resistance. (More on that later)

Tauro:
I took out the C3420 transistor in the middle and tested its function with my multimeter, and it seemed fine out of circuit, then I soldered it back in.
If I'm getting the pins in the right order, when there is power to the motherboard it's showing a bit over 4 volts on the emitter, a bit over 5 volts on the collector, and a bit over 3 volts on the base. No idea if this is normal.
With the transistor in circuit and powered off, I'm getting continuity (or close to it) on the emitter and collector with 5v, and that weird low 50 ohm resistence on those pins with ground.
If that transistor is not the problem, where might this low resistance be coming from? A fried IC? A dead cap or resistor?
Based on no real evidence whatsoever, I'm beginning to suspect the clock generator chip... problem is, it doesn't seem to be documented anywhere.

Stamasd:
I don't have an oscilloscope unfortunately. I do have a logic probe if that helps...? The crystal is getting power, and as far as I can tell it's producing a signal. I'll see if I can chase it around the board.

Also, in regards to the cache chips - I assumed they were real because they are ceramic and socketed (all the fake ones I've heard of were plastic and soldered right onto the board) and the lettering doesn't come off with alcohol. But if they are fake, at least that's one variable I can eliminate.
So I'm happy to take them out for now. Just not sure how to set the jumpers - all of the documentation seems to expect 128kb minimum.

Quick update: don't know why I didn't check this earlier. With the motherboard powered on, there is no longer any continuity/low resistance between 5v and ground.
Again, I'm still a n00b at electronics. No idea if this means anything...

How good is your multimeter?
If it's very good and shows miliohms, you can find the general area and then test components in circuit for continuity.

If it's not that good you're not going to get great results. I'd suspect caps or transistors first.

Isoproply can be a way to spot heat you can't feel.
It will evaporate around the short first.
Otherwise... Pull them one by one and see if it changes.

Figure out the short first before chasing the clock.
It's never going to work with a dead short on 5v.
I'm surprised the psu isn't cutting out with a dead short.

Oh. You posted.
Hmm. That seems weird. Shorts don't normally resolve that way.
Are you getting 5v at the normal points?

Take a look at jp5.
In your pic it's open.
The manual suggests that it is a voltage select for the bios.
Google the datasheet for the bios.
If it's a 5v chip, set 2-3, 12v 1-2.

Then see if it lives.

Check all the others while you're at it.
https://theretroweb.com/motherboards/s/pcchips-m918-486#docs

I've got a reasonable multimeter. I don't think it goes down to milliohms though. (It's a digitech QM1523). The lowest ohm value on the dial is 200.
As far as I can tell, when the mobo is powered on, I'm getting 5v everywhere I'm supposed to. But I'll try the isopropyl trick and get back to you!

I can't find ANY info on the BIOS chip online. I'm pretty sure it's a rebrand of some 27c010-ish thing. Not sure what voltage it takes, but probably 5. Changing j5 made no difference.
I've also tried various jumper configurations with various ROM images for this board on various different chips. Nada.
In this thread:
[SOLVED] - BIOS issue with my PcChips M918i Motherboard
I got the impression that j5 is just programming voltage.

Oh yeah, turns out those cache chips ARE plastic (I scraped a bit off one corner of one). My partner, whose hobby is pottery, will be very disappointed to hear I can't tell the difference between ceramics and plastic! 😁

Small update:
It doesn't seem like anything on the motherboard is getting particularly hot.
Weirdly, the first place the alcohol always evaporates from is the UM8663AF chip. The one for I/O. What that chip might be doing without a clock signal is beyond me. After a couple of minutes of operation it has warmed up slightly, but not heaps.
Second place goes to the M1489 chip, then a couple of other little ICs here and there.
Notably the BIOS chip and the clock generator stay cool for the longest.