These mobos did suffer from capacitor plague, did you replace the ones near CPU slot? If not take the mobo out, the CPU out of socket, and test the caps for shorts. Chances are one of the CPU power rail caps has shorted, took out a MOSFET, which in turn killed the PSU. It's a theory but it's not impossible.

If it's easier, and the nothing is smoking, you can probe the running mobo for voltages present. See if all the voltage regulators (esp. near CPU, but also RAM, etc) are doing what they should. If you do get correct voltages then you need either a POST card or another CPU to test with. Preferably both to be sure.

Knowing it's an ABIT, replace caps first. I've seen them fail with zero outside signs.

I'd heard about the capacitor issues with these boards but it felt like too easy of an answer for getting the board working again. Deunan's theory of something on the board shorting out the power supply is definitely interesting though. I'll check the caps and the voltage regulators and report back, and I'll look into getting a POST card. Thanks!

Any ideas on how to free the CPU module without those retaining clip levers being there?

Pictures could help.

Well I went to take photos of the CPU but ended up being able to get it free fairly easily right afterward haha. All it took was a bit of gentle flexing on the edge of the slot.

I also took the time to tone out the power supply connector on the motherboard, and I did find some interesting things. There appears to be a dead short between the +5VSB pin and ground, and only 50Ω between the 5 V rail and ground. It looks like the -12 V pin is also bent a bit, which may have happened when I swapped the power supplies; the plug for the new one was noticably more difficult to connect snugly. I'll definitely repair this, but I doubt it was the only thing preventing the computer from booting. So I will pull the board probably this weekend and investigate further. Thanks to everyone for your help so far.

What is the brand and model of the new power supply? Has it been verified working with other known working hardware?

If you suspect a short somewhere it'd most likely be a capacitor or the CPU voltage regulator.

I don't usually see dead power supplies from a board that has died (or when I have an oopsie) though, short circuit protection usually does it's job. Usually it's one thing or another, not both, but anything is possible.

Did the old PSU have a -12v rail? Does the new one have one? This is not a given - I'm not sure when exactly this requirement was dropped from motherboards, I do know that some wont work without it, and unfortunately some later PSU are missing it. Some without it I've opened up just to find the circuitry in place, but without an actual wire attached (must have saved a whole 1 cent there).

Post pictures of the motherboard when you get a chance, as the other have suggested here.
If the motherboard really does have a true short-circuit between 5VSB and ground, it's possible the old PSU may have gone over-voltage on the 5VSB and killed something.

That said, do NOT toss the old Topower PSU in the trash just yet. If anything, we need to confirm if the PSU 5VSB circuit is indeed what killed the motherboard or not. Many old ATX PSUs used what's known as a 2-transistor self-oscillating circuit for the 5VSB, and some of these have been known for going out of control / over-voltage over time and killing things (e.g. the Bestec ATX-250-12E is one of the best-known examples in HP PCs).

So if you have a multimeter, take that old Topower PSU and measure the votage between the 5VSB pin on the ATX connector and ground. You should get a nice steady 5V +/-0.25V (e.g. 4.75-5.25V is acceptable... though I would expect much tighter regulation usually - something like 5 to 5.15V).
If you get nothing on the 5VSB pin (or very low voltage) or voltage that is too high and out of spec, then unplug the PSU, wait for like a minute, and them open it. After this, take good pictures of all sides under good light and post them here. That should tell a much fuller picture about what happened. If the PSU really went over-voltage on the 5VSB and killed something on the mobo, chances are the SIO/LPC probably bit the dust and you'll be unlikely to save it. But these are pretty rare cases.
IMO, also double-check all of your resistance measurements on the mobo and certainly do them with the CMOS battery removed. Moreover, if the new PSU can supply 5VSB voltage without the mobo tripping and killing it, then the short-circuit you measured is likely a fluke. So keep that in mind.
Bad caps could also be the issue, as the others have suggested. So we have to tackle this from multiple sides to try and figure out what happened... which is why I also think it's a good idea to see the original PSU inside.