All things being equal, plugging into a 120 volt source is all you need to run the fridge.
However, nothing ever is that simple.
As stated previously, in order for shore power to run the fridge there has to be a solid 12 volts of DC power available.
Normally, the converter will supply that 12 volt control voltage. If the converter is (broken/disconnected/just not getting to the camper due to an open fuse), the battery should supply the 12 volt control power until it is dead (or too low).
If the battery is dead (voltage below 11 volts), even if you FIX the converter, it may not turn on. There is circuitry in the converter to prevent you from powering up a "shorted" or "dead" battery.
That is why you ALWAYS disconnect the battery to troubleshoot the converter.
Lou and Laura with Bella - German Short Hair Pointer
2008 GMC Sierra 2500HD Crewcab SB Allison Duramax
2010 Flagstaff 8526RLWS - Superglide 3300
HAM CALLSIGN - KC3FFW