That sounds like the input cap but I would look at a print to be sure, that is if you can find the print. You will have to test the zener diode to make sure and if you're reading with a standard DVM now on ohms scale Im almost sure its shorted OR whatever else is in parallel with that diode itself. If the transistor is getting hot its being driven into conduction but is it a square wave making it up the leg or something feeding it DC? A square wave means something is trying to work and I would venture to guess a cap is soaking everything up through ESR leakage. Most consumer grade switching supplies are of the single ended HEXFET design or IGBT transistors, they regulate by controlling the duty cycle of a square wave around 100 KHz. If that transistor was shorted the fuse would have blown as the transformer itself is only a fraction of a DC ohm but several hundred ohms at 100 KHz, the transistor may in fact be leaky and given the heat stress I put money on that as well. Is there an IC on the board marked uPC 494? These are notorious for failing without provication.
The zener may be a shunt regulator to protect the gate/source of the FET on the heat sink so check the zener AND the FET out of the circuit using the diode check function of your meter if you have it. A good FET will show around 0.600 volts forward between gate/source and will show open in reverse. Also check ALL the caps on the board along with the opto-isolator, that is the square IC with only 3 ot 4 pins coming out of it and it may be white colored. The caps may test good using an ohmmeter but dont be fooled as the leakage may rise with 100 KHz of rectified squarewave coming at them. If they get leaky enough they can saturate a power supply to the point of foldover leaving you with a very smelly supply and little output if they havnt exploded yet.

That should get you started and I will watch this thread for a while and offer you a hand grenade to finish the job!

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