Note that failures generally cascade. A typical failure starts with an overheating MOSFET, which fails as three-way short, dumping a fair fraction of the DC bus cap plus mains fault current into the current sense resistor, and gate circuit. Both flow into the controller, which is subsequently damaged. Resistors may also fail (shunt, or the resistor between it and ISENSE, or gate resistor and whatever else is with it (often a parallel diode for fast turn-off)).
The fault current blows the fuse as a final act, which might be tempting for its simplicity / obviousness -- it might be the first thing you check, but replacing it is the last thing you need to do in this case. (Rarely, the fuse blows by itself, perhaps due to fault manufacture, thermal cycling, or mains surge; highly suspicious, but can happen.)
Startup resistor going open is relatively inexplicable, but perhaps an overvoltage condition occurred, or it was a thin-film type subject to environmental effects (corrosion).
A damaged chip may continue to work, in part. A typical case is the output devices or ESD clamp diodes failing shorted, resulting in excessive current consumption, or excessive input loading, or insufficient output voltage. This could be the case here, where the chip's UVLO/startup circuit is working fine, but when it enables power to the rest of the chip, it tanks the supply. It could also be more direct, such as gate drive dumping into a shorted MOSFET -- you didn't mention if this was replaced or checked so as far as we know, it is still an option.
Also, because of ESD clamp diodes, it may be that the aux supply got overvolted, which could damage that capacitor for example.
More likely the aux cap just dries out, as it's a fairly stressed part, actually, despite the fairly low output current -- the controller typically needs a couple 10s of mA to run -- but it's also a small capacitor, and ratings in small sizes are rather modest.
Secondary side can end up toasted too; such a transient can dump excessive current into the rectifier, or maybe the failure was induced by a partial or complete failure of output filter cap or rectifier, or regulator (TL431?) for that matter. And then the primary side tries to start up into a partial or complete short, and keeps chugging like that forever, or until something else blows up.
Have also seen one where the controller was just dead. No idea how. Everything else checked out, replaced it, worked fine. Go figure...
In short, replace everything active (transistor, controller, some diodes even), and check everything passive (resistors and capacitors). Replacing electrolytics wholesale isn't a bad idea, but generally isn't necessary as long as they test OK.
Tim