It is obvious that you have very little experience with troubleshooting and repairing any electronics, much less modern PCBs with surface mounted devices.
The classic method of troubleshooting is to use an oscilloscope to tract the signal through the circuit. This requires a schematic so you know what that circuit is and a parts layout diagram so you can find each part. But a well labeled PCB can be substituted for the layout diagram. If the schematic has waveforms on it, that is a great help.
But I doubt that you have that. So what can you do?
1. Look for obvious problems: bad solder joints, parts that are burnt, other visible faults. A good magnifier or low powered microscope is a good tool here.
2. Look for bad electrolytic capacitors. They often have an end that is bulging. If you see any like that, replace them. In fact, if some electrolytic capacitors show this, replace all the electrolytic capacitors because the others will often fail soon anyway.
3. Look for other obvious faults. Bad switches, which usually can be tested in circuit. Burnt relays. Smoked inductors. Anything obvious.
4. If you haven't succeeded in finding the bad part(s) by this point and you do not have a schematic or a scope, then your best and easiest path is probably just replacing all the ICs and semiconductors (transistors, diodes, etc). Active devices fail much more often than the passive ones. I have had to do this in spite of years of experience and having all the above mentioned items at hand.
5. It is hard to measure resistor values or the parameters of other devices while they are in the circuit. This is because an Ohm meter or other tester will "see" not only the part you want to test, but every thing else that is connected to it. So, for instance, a high value resistor can give a very low resistance reading because of other paths through the circuit it is connected to. The rule of thumb is that all the leads, except for one, should be completely disconnected. With surface mounted parts, this usually means ALL the leads.
PS: Yes, resistors almost never fail by having their resistance fall to a lower value. Most bad resistors I have seen were obviously burnt up and showed either a very high resistance or an open circuit. And I seriously doubt that your problem is a bad resistor. That would be my very last guess.
PPS: If there aren't any bad electrolytic capacitors on the board, then that "black thing with the heat sync" is probably where the fault is. And, if that is so, then the board and the switch is trash - er, that is, not economically repairable.
Good luck!