Well, the U2 threshold stuff is just wrong, but it doesn't affect anything and you can ignore the LEDs.
VR2 should be a fixed resistor, and the variable part should instead be in series with R32, but at least with this type of converter (as opposed to a one-switch flyback, say), output voltage is limited by turns ratio, so it shouldn't self destruct when VR2 is set wrong.
I like that it actually has current sense, inductor current sense at that. It's not a 100% average current mode controller, but it seems it was compensated to operate smoothly under current limit so that's good.
C14 and C15 will have a hard time, but I suppose that doesn't matter. Don't load them heavily.
No polarities are shown on the transformers.
There could stand to be some R+C dampening on the output rectifiers, depending on transformer design. Those are usually seen in ATX supplies.
Oh, F1 is way too fucking big, haha.
I don't get why all the junction diodes are drawn as upside down schottky (except D9, D17), and D15 isn't [ed: oh, but it's a PN junction diode, and not 30A; weird, nevermind], but whatever, no accounting for style...
Weird...
SS9013 is the original, well known, small signal BJT of that number (don't remember who created it)...
2SC9013 is almost certainly not a proper Japanese (EIAJ?) designation, but a co-opting of the numbering scheme? I see a datasheet by an Indian manufacturer... it's a transistor for "potable radios". I'll, uh, stick with the bottled water, thanks.
(Yes, any general purpose NPN will work, I've seen plenty of 2SC1815s in that position.)
Anyway, without analyzing it too closely, it should be okay.
As for faults, yeah, both transistors need to be replaced at the same time, and probably a lot more, as you've discovered. Likely mechanism, fault current flowed through both transformers, delivering enough surge current to burn out transistors. I would guess cause to be accidental reverse polarity on the output, or maybe an input surge.
Tim