Hey Magic, thanks for the insight here. Your last statement makes sense now. I couldn't quite understand what I was seeing by changing that cap.
I did check that feedback path and everything matches the samples the company gave me. I looked for tombstoned resistors and everything, no dice. I guess something is wrong with the auxiliary transformer winding . . . but everything I did matches the samples the designer sent me . . .
Maybe the feedback resistors are swapped, then? I'm not sure what else could limit the output voltage to 4.5V for the entire time when the controller appears to be operating (discharging its supply cap). But as long as the secondary voltage is clamped near 5V, the auxilliary will also be way lower than normal.
The idea above is not bad: apply 5V or 12V to the output, verify if TL431 reference input is less than 2.5V, check if its output is high, check if there is any excess current through R32 that doesn't flow through R37. Maybe you missed something, maybe the chip is blown, maybe it's TL432, maybe you screwed something up while transferring their design to your board and still can't see it.
BTW, R37 seems a bit high in value because it only passes 100µA but the TL431 may sink 400µA even if REF is less than 2.5V. You could bodge an extra 1kΩ on top of R37 and see if it helps.
The reason I think something is wrong near TL431 is because a voltage like 3.5~4.5V is the minimum for such TL431+opto circuits to still sink any current through the opto. It looks like as soon as this minimum supply threshold is reached (and no earlier), the opto lights up and throttles the PWM.
Another thing I could theoretically imagine is some semiconductor or insulation breaking down and clamping transformer voltage, but then the PWM would run at full throttle and pump several watts of power into the transformer and all those watts would be getting dissipated somewhere.