Generally yes- you are correct. Most campers the jacks are powered/connected directly to battery(s). They dont run thru inverter/charger- but most will run better as your simply replacing or offsetting what the jacks are drawing-
Why when one battery removed they don't work is a puzzle-but many variables or coincidental issues that might be contributing and not considered. Why trying with charge source and one battery has no effect is even more a puzzle. My first thought is just not enough reserve- if 'test' was done after removing the bad battery likey the other battery was drained. Trying to think how battery wireing could /might cause this but cant see it. Dont know what 'big boy' jacks are, cant imaging they'd draw enough amps to stall, at least briefly should work. Meter on the battery might be interesting. Curious