Actually troubleshooting it could be a challenge, hard to say without a closer inspection. The overall design does look roughly ATX-like.
Those coils look to me like saturable reactors, maybe -- these are sometimes used to ease the hard switching of rectifier diodes (more common back in the day, when diodes sucked more), or control the output voltage (rarely used for main supplies, but surprisingly common for supplementary supplies -- the average ATX unit uses one to derive the 3.3V rail from 5V windings).
Whatever they are, it's unlikely they're actually ferrite (black ceramic) -- that material isn't usually used in toroidal inductors, and doesn't usually have enough losses to fully nuke itself. Metallic cores (ranging from powdered iron to metallic glasses!) are useful for inductors and saturable reactors, but they're more lossy, and in particular, are prone to runaway failure, where time*temp causes breakdown of the material, causing increased losses, causing... well, you know.
Or it could be good old fashioned cooking under load. But this seems unlikely, given that the coils behind them are even larger, and likely carrying full load current, and don't seem to be suffering at all. That's why I think they're doing something active, like diode switching, or voltage control.
The transformer looks okay: the tinge of darkness looks like ordinary transformer varnish. Not that it couldn't be nasty inside, of course.
Tim