well, that eleiminates the thermostat as the source of the trouble, now it is pointing towards the compressor & it's overtemp unit, ( a little black disk with two wires going to it) this is the "click" you hear every 10-20 seconds, it acks like a circut breaker, as the compressor trys to start, all of the current first goes through this overtemp breaker, if the compressor doesn't start to turn, then the overtemp breaker senses the current rapidly rising ( a non-running or "locked-rotor" electric motor will draw lots more current than a running motor) & there is a little disk of metal that will start to overheat & when it does, it pops open the circut & you'll hear a "click" (just like a real circut breaker for your house) but after it cools down ( about 30 or so seconds) it will reset & "click" again, this time reconnecting the compressor to the line voltage.
now it could be a few things, 1.) a weak overtemp breaker, 2.) a non functioning start-run relay, ( this is located mounted on the compressor usually right were the wires go under a metal or plastic cover on the side of the compressor, or on older machines it's off to the side in a seperate metal box & only three wires goe to the compressor directly) or 3.) a bad stuck, or locked/seized compressor.
you can try a 3-in-1 hard start booster relay to see if the compressor will still turn over. this would replace the original overtemp/start-run relay circutry attached to your machines compressor. but it's not a cure-all. it may help your compressor to start up if it's a little tight or a piece of sludge has gotten the piston sticky & is cousing it to have too much friction on startup than the electric motor inside the compressor can overcome normally) or it must likely is time for a new compressor.
if the overtemp breaker is accessable, you can test it by TEMPORARLY jumpering across is & see if the compressor will turn over, but don't Try for more about five seconds, otherwise the compressor will over heat internally & you could fry the wiring. IF IT DOESN'T START RIGHT AWAY WITHIN 5 SECONDS.....STOP.....& DISCONNECT THE POWER
& don't try it again....you'll have to let the compressor cool down for about 30 minutes, then once you get a 3-in-1 hard start booster, you can try that. but in all likelyhood, it's the compressor if the 3-in-1 fails. remember, "stuff" happens & you are dealing with a +40 year old machine....