You can certainly come up with a reason to have an isolation transformer. But for most things, it's just a false sense of safety/security that will let you convince yourself that it is reasonable to do VERY UNSAFE things.
What matters is the DIFFERENTIAL voltage between two points. If you float the mains, you still have that differential and have gained nothing. To be safe, you still have to put both hands in your pockets and leave them there.
So, you're bored standing there with your hands in your pockets. Let's do some unsafe things! Let's make a measurement with the oscilloscope. So, you grab the probe. Hey, what's this black wire dangling from the probe? Let's hook it "here" in the circuit. Doesn't matter where "here" is, you've now got a NOT FLOATING power supply. What's worse, the normally isolated secondary side may have a lot of common-mode volts on it. You're at risk of a shock...AND...simultaneously blowing up your scope and anything you're using for a load.
Well, we can't have that. Let's cut the ground pin on the scope power cord. That'll fix it...wonder if they have WiFi in heaven so I can report my progress???
If you want to work on power supplies, get yourself an isolated scope probe. Or a portable scope with no metal parts designed for that type of measurement.
Tektronix A6902 probe works well. I picked up mine at a garage sale for a buck. I suspect they're substantially more from a dealer.
Using a DVM with well-insulated probes, you can tell if the diodes are open or input storage cap is defective or there's volts on the fet. Beyond that, you really need more than an isolation transformer if you expect to do it safely.
mike