“I have heard from a frum repairman that most modern fridges use a timed cycle for the compressor as opposed to the older models that a rise in temperature caused the compressor to go on.”
Frost-free refrigerator/freezers use both one or more thermostats as well as a timer. The timer periodically shuts down the cooling mechanism and activates a heating unit installed by the fins of the condenser coil in the freezer cavity to counter-act ice build-up. It then reverts the appliance back to normal operation. However, these refrigerator/freezers most certainly do depend on thermostats to monitor/maintain the selected temperature during normal operation.
For those interested, there are some very helpful appliance repair videos on-line for lay-people explaining what these components do and how they are replaced when defective.