Great find. Looks like a TDD-1? I'm not a refrigeration expert by I have had to learn a bit about the TRUE commercial units as I have two GDM (Glass Door Merchandiser) units. I would suspect that the TRUE kegerator units function similarly. Yes, the compressor and condensor fan should cycle on and off together. The evaporator fan inside should always run. Does it appear that anything has been modified in the unit? If you suspect a bad thermostat, research the replacement thoroughly.
My TRUE GDMs use a thermostat called a Constant Cut-In Control, that monitors the evaporator internal temperature, NOT the air temperature of the interior, which is a fundamental difference for a thermostat. The temperature probe is inserted into a hollow tube built into the evaporator. The thermostat operates with evaporator temperatures that drop to well below freezing during normal operation. Once the evaporator temperature is reached, the compressor is turned off and it will not "cut-in" until a specific temperature, well above freezing, is reached at the probe. This is all done in order to provide a defrost for every cooling cycle and does not allow short cycling of the compressor to occur, which allows these commercial coolers to run trouble-free for years.
When obtained, my GDM-12 had been modified with an air sensing thermostat in an attempt to use it as a ferm chamber and not having the original, I had to research a replacement. I ended up installing a Ranco A12-701 thermostat with a 23.5deg Normal OFF / 41deg Cut-In temperature and it runs like a dream as a corny cooler.
Good Luck