Time to see if my initial suspicions were correct by examining the circuit board.
Before you ask, no, the components around the 5VSB feedback and main output pulse-width modulator chip are not burnt. The darkened area is just shadows. I took a few PCB shots during the repair, but none of them turned out good enough to use here due to perspective, framing, focus or other issues, so I had to re-open the supply once more to take this one.
The 5VSB output is provided by an all-discrete flyback circuit and a transformer with at least one auxiliary output based on the five wired pins on the transformer's primary side. The main chip of interest is the ST-branded switching IC that ends in “44B”. Its other markings appear to have been burnt off by the glue securing the nearby polyester caps. A quick online search for DIP-8 flyback regulators returns the UC3844B as the most likely match, though that remains to be confirmed by cross-checking the pin-out. Other integrated circuits of lesser interest, at least for this initial assessment, include a TI TPS3510 for power supervision (explaining the shutdowns roughly 80ms after outputs fail to ramp up), three Lite-On 817CN photocouplers and four Unisonic (UTC) TL431 programmable shunt voltage references.
At a glance, all capacitors that appear to be related to the feedback loop and switching circuit operation are either ceramic, polyester or other metal film for high stability and reliability, so the likelihood of a capacitor failure in those low-current circuits should be unlikely.
The TL431 chips (TO-92 packages) located directly next to photocouplers are obviously controlling their respective feedback couplers. The other two are not so obvious from the top but still guessable based on what else is nearby: the one located next to the orange wires is used to tie the ATX cable's 3.3V sense wire into the PWM feedback loop, while the other, located next to the fan header, manages fan speed with the help of two thermistors (one near the heat sink and the other at the main transformer).
Looking at the 5VSB's primary side area, it is almost entirely self-contained save for two traces sticking out of it: one to bring power from the top input filter cap in and the other connecting to the PWM controller area, presumably to power it up via an auxiliary output. This auxiliary output has a 47µF 25V capacitor on it, making it a potentially interesting target for analysis.
The middle of the primary side is used by the main MOSFET's (W12NK80) gate drive, snubber and current-sensing circuitry. If anything went wrong there, magic smoke would get released in no time flat, so there is nothing worth messing around with.
Following traces and components starting from the main MOSFET back towards the pulse modulator reveals a match for the gate drive and current-sense pins. Doing the same from the bottom input filter cap through jumper wires also reveals a match for the GND pin. Tracing the path from the 5VSB's auxiliary output also confirms a match for the VCC pin, and that the PWM is powered entirely by the 5VSB's auxiliary output. Components attached to the remaining pins are consistent with feedback, compensation and timing functions. Short of reverse-engineering component values, UC3844B is an exact match. The 22µF 50V bypass capacitor across the PWM circuit's VCC and GND pins would be another point of interest.
Have you ever wondered what sort of path the PS_ON signal takes from the ATX wire to control the main outputs? If the answer is yes, follow the red line. It starts with a thin trace breaking off from the 5VSB output, sneaking behind the main transformer output to reach the photocoupler that controls the main switcher's power, then creeping all the way around the board to reach the TI chip. That component is also powered from 5VSB. The trace goes around the PCB's entire secondary side.