Yes beware. Make sure nothing's easily flammable in the room too, because the things it projects could be fire hazard I think.

To test thing you can simply disconnect one leg of each tantalum capacitors. The board will probably work fine without it and you'll see if a tantalum was the culprit.

maestro wrote on 2021-07-10, 03:45:

Ever see the movie Sunshine? Look at my values ... What do you see? *Epic music commences*

Can't say I have. Or I don't remember it. But point is, there aren't any obvious shorts. The varying results are due to various capacitors charging up a bit, that's normal.
So, perhaps it's not a capacitor issue after all? Remove the CPU, try again. Obviously the mobo won't boot but you'll see if the PSU is now kicking in properly or not. If so it would point to the CPU, it's voltage regulator, or the PSU just not working properly (mobo by itself draws little power).

Could be PSU caps - shorted or dried out. Tests for shorts, obviously with PSU disconnected and prefarbly give it a minute to fully discharge so you don't damage your DMM. Test resistance between each colored wire and black (which is GND). Some lines might be as low as 10-15 ohms, especially the 5V one, but less than that would suggest a shorted capacitor. Inspect it visually as well for any signs of leaking electrolyte or bulging/venting.

maestro wrote on 2021-07-10, 02:41:

AT P8 (left) varies* 0.3828 k varies* varies* 0.23 gnd-gnd 0.30 gnd-gnd […]
Show full quote

AT P8 (left)
varies*
0.3828 k
varies*
varies*
0.23 gnd-gnd
0.30 gnd-gnd

AT P9 (right)
0.14 gnd-gnd
0.15 gnd-gnd
open circuit
0.3827 k
0.3827 k
0.3827 k

These readings look fine. You problem doesn't seem to be a tantalum on a power rail, but something more difficult, like a short circuit that is enabled only in certain logic conditions. If your mainboard is auto-voltage (i.e. it doesn't have a jumper to control 5V/3.3V, but it does support 3.3V at all), the auto-voltage logic might be toast. Also, on auto-voltage boards, you don't see a shorted CPU at the AT power connector unless multiple voltages are properly asserted at the same time. Try to test for a short in the CPU voltage circuit by measuring CPU VCC (check a 486 pinout diagram) to GND.