So is the PSU an external brick ? IDK the PS voltages, but I'd guess whatever enters the PS, would power a bunch of lower voltage rails, like 5V, 3.3V, 2.5, 1.7, stuff like that for a bunch of chips.
I'd start at the input, and try and find the main power sections, there's probably a standby rail too. Go around and find big inductors, most of them would very near a controller chip, making some voltages rail. With any luck, at least the power controller chips, would be labelled and have public datasheets. I'd lookup as many chips as I could, and on the PCB's see which ones are getting power.
There'd be linear regulators too, like 3 terminal ones, like for analog and digital voltages rails, on stuff, it's common to see 5V made with a DC-DC SMPS, and another 5V rail from a linear regulator.
The rails should power up in a certain order too. PC motherboards and GPU's they just use a few AND gates and ENABLE pins and Power-Good pins on chips, to do that.
If you were lucky, the problem is just in the power sections. I'd find that a lot easier to track down that some digital problem.
Maybe it's just the switches/buttons are dirty too.
I'd recommend soldering on a few wires, it's super easy to touch the wrong thing with the probes, or have them slip. But I find even that is a nightmare on mobo's, I've damaged traces just trying to get test wires on there.