A/c not blowing cold. Brought to the vehicle to my mechanic. They advised it took numerous hours to diagnose and that I would need to replace the A/C amplifier. Amplifier was replaced they called and said ac still not working and that hte compressor was shorted out my likely by the a/c amplifier. Is it possible the amplifier was fine all along and the compressor was bad?

 

Sounds like you need a better mechanic who can do a proper diagnosis. They should be able to test the removed amplifier.

I never heard of this part and a quick search found this general information on another Toyota model

The chassis manual has a nice section on the AC system, even diagrams the circuit for you. the "magic blue ****" does exactly as mentioned, adjusts when the amp will let the clutch engage in regards to RPMs.
Clockwise=lower RPM
CCW=higher rpm required to run the AC.
The amp gets a signal from the coil, same wire as the tach, that it reads for RPM.
Thermister and pressure switch functions are also controlled in the AMP, it all come down to telling the compressor when to run and when to shut off.