It seems very complicated to detect the problem of this module...
Depends with what you compare. Of course it is not the simplest thing in the world. But also it does not look to me, that you spent a lot of time on this problem. You measured that there is nothing at the output side, capacitors are good, rails not shorted, output mosfets good. You also measured that main PWM controller is not powering. And that is pretty much all as far as I understand. Have you traced how the PNP transistor is controlled? Mains undervoltage circuit?
A lot of people in this forum would likely troubleshoot such board in one hour or less, but they will not be able to tell exactly how to do it, because there is no exact algorithm, it is just probing, thinking, reverse engineering sections of PCB, finding some datasheets, drawing schematic sketches on a piece of paper and probing again until something is found and the problem understood.
Also I will note, that people generally hesitate to offer ideas which might be risky. For example in this case you might consider to force start the SG3525 by bypassing the circuit which controlls 15V. But it is risky, bad things might happen, more damage to the board. So if I suggest this idea and you respond that your SMPS went on fire and now it is totally destroyed, I would feel really bad about it.
You might consider such repair as opportunity to learn. It really depends what is your goal. If you want just to save some money, then maybe it is not worth.