Check with an appliance parts supplier to determine whether the Ranco unit is a specified replacement for the Beaumark. Also, it is possible that the Ranco thermostat is defective, though new.
Clicking sound is overheat at the compressor. There is a device called a Clixon that is cutting and restarting. Remove covers and clean the condenser grill with a vacuum cleaner and brush.
Could you comment below with more detailed photos of the back of the orange switch and any other bits with it. If the switch isn't broken It might just 'click' back in. I don't know without better images.
That unit is surprisingly clean. It may have had the fan module that you can see in picture, replaced and whomever replaced it, did not put the lug back on or it was not there with the newer unit.