Is it safe for me to assume the transistor needs to be replaced or should I continue to resolder everything and gather voltages? Or should I desolder two pins from all the transistors and test them in the same manner?
Put it back and start testing the voltages. Nothing disastrous will happen even if the transistor is faulty, and the voltages will tell the story.
The very first voltages you want to test are the supply voltages on the PCB, to make sure that the +13/-13V you think you have applied has actually been applied.
If you cannot find components, like some of those capacitors, you can work out where they are by looking at the circuit. If they connect to a switch and are not on the PCB overlay, they will either be mounted on the switch, or they will connect somehow to the wire that comes off that contact on the switch. Just follow the wires and connected PCB traces and you will find it.
As I said before, the schematic tells you many of the voltages that should be there. If you are not sure about which lead is which on a transistor, look at which other components a lead connects to on the PCB, and then compare it with the schematic. So of a transistor lead connects to R123, and R123 connects to the base of the transistor in the schematic, then you have found the transistor base lead.
I think that 2N3902 was in the Automatic Gain Control (AGC) circuit. The job of that circuit is to adjust the gain in the oscillator circuit to exactly 1.000 . If it is exactly 1, you have a beautiful stable low distortion sinewave. If it is more then 1 , you have a badly distorted sinewave. If it is less then one, you have no oscillation. The best way to see what is happening in the AGC is to measure voltages.
Richard