No one directly answered your question about what the batteries are used for.
First, and most important from a towing safety perspective, the onboard batteries provide power for the breakaway switch that will apply the brakes if the trailer comes unhooked from your tow vehicle. This is a fairly unlikely scenario, but would put you in a bad liability position if it happened and the trailer did any damage or injured someone.
The battery provides power for the whole 12v
system when you're not plugged into campground power. Depending on the age of your refrigerator, it may require 12v
power for the control board even if you're using propane or 120v power to actually chill the refrigerator. The 12v
system also powers all of the lights inside your trailer, the water pump that's used if you're using the freshwater tank instead of a campground connection, any vent or exhaust fans built into the trailer and (as you learned already) the electric jack.
If you plugged the trailer in and the jack worked again, and if the interior lights work when the trailer is plugged in but not otherwise, then the Univolt (or some replacement converter-charger) is working for now and it'll work on shore power. As already mentioned, the original converters weren't designed to work properly without a battery in the system, and you may get higher voltage than you expect on the 12v system, or varying voltage.
For the breakaway system alone, batteries are essential to have to tow legally.