Very true - That is if for some reason unknown to us, some electrical expert (so to speak) changed something up. I once worked on a boat that lost power to the microwave. Owner was not electrically inclined at all. After removing the microwave I found that the receptacle it should have been plugged in to was empty and the microwave wire went through a hole to the left. The dedicated receptacle had no power. So I found the end of the microwave power cord. It had been cut off, spliced with wire nuts to another orange extension cord and was plugged into the outlet in the head. Yes - the GFI. The original outlet black feed wire had come off due to a loose connection which rendered it dead and also a safety issue. Ya just never know. Follow the circuit. Start where you have a full power reading and stop at every switch, plug, junction and test you are getting the same power (no significant voltage or amperage drops) on both sides.
One quick way you can gain some information is to plug your fridges 120vac into a non boat power supply such as a cord directly from an electrical source you know is patent (pier). This will tell you if it is your boats power (or lack of) or if it is the fridges issue. You can also try other appliances (i.e. electric drill, wet dry vac - something that pulls substantial amps) in the plug that your fridge would normally plug in to. This would also help to let you know where your failure might be. Thee are of course meters for all this testing, but you really just need to figure where your problem is - boat or appliance. That at least narrows it down.
For the dc side of this, I am with others on this post, your batteries are toast. Going down to 10.8 volts is non recoverable. It is not enough to power up the fridge compressors and hopefully they didn't burn up continually trying to start up. Aside from your windless, bowthruster (if you have one) and the like appliances, refrigeration is one if not the highest draw on your boats 12vdc system.