Pretty much comes down to 3 items:
1).Defrost heater
2). Defrost limit switch - Usually a bi-metal switch in series with the heater and shuts it down if it goes over a certain temperature.
3). Defrost timer - The "brains" of the refrigerator. Tells the temperature switch when it is allowed to tell the compressor to run. Takes the temperature switch's control away (every 8, 10, or 12 hours depending on the setup) and tells the defrost heater to work for 20 - 30 minutes (depending on setup). Then gives control back to the temperature switch after the defrost cycle.
Slim chance but, still possible, is an ice bullet in the freon line or a burnt wire.
I don't know how anyone could know which is the problem child without going to where the refrigerator is and testing it. I have over 25 years experience repairing refrigerators. If you are handy with a multimeter and mechanically inclined, it is a repair you could do yourself. The repair place isn't really steering you wrong, in my opinion.