Power failures and on and off of power can be a real issue.
This on and of actions is really hard on relays,overload devices on compressors. Sometimes it can cause overloads relays to over heat,contacts to burn and prevent them from coming back on or from functioning properly. Its a common issue with many of the new refrigerators.
With today appliances you really can't just make a judgement call on what is the real problem without proper testing. They have defrost timers,thermostats,overload relays,fans,defrost coils,start and run capacitors and anyone of those things can keep the refrigerator,compressor from functioning properly. I doubt it is a serious problem.
It will be costly. Repairmen are not cheap.