In trying to trouble-shoot this problem myself (at least to see if I could find the root cause), I confirmed that the normal voltage existed across the wires from the thermostat, but discovered that at least one of the two "pressure switches" has an open circuit -- these "pressure switches" are wired in series, between one of the thermostat wires and the compressor control unit. I think that each of these "pressure switches" monitors the refrigerant pressure (one pressure switch on each of the refrigerant lines leading back to the house), and is supposed to interrupt the thermostat voltage if the refrigerant pressure gets too high.
I seem to remember, for example, that if you run the A/C, then turn it off (by raising the thermostat setting), then immediately turn the A/C back on (by lowering the setting), there is a chance of "blowing out the compressor," since when it comes back on, the pressure it's pumping against is still very high. So, probably a main reason for the "pressure switches" is to prevent the compressor from starting up if the refrigerant pressure is still too high from recent pumping.
Anyway, to see if the problem really was just one of these "pressure switches," I shorted the two of them out. The compressor started like a charm, and has been running perfectly, cooling the house.
My question: If I continue to run the A/C system this way, with the pressure switched shorted out, until I can get them replaced, am I running a major risk? Should everything be OK as long as I'm careful not to re-start up the A/C right after it has been turned off? I.e., if I'm careful to wait 5-10 min before re-starting the A/C (or just let it cycle normally via a fixed thermostat setting) is there no danger of "blowing out the compressor?"