So, the idea here is that when someone points out a fallacy in an argument, the argument must be wrong... right?
Wrong. Just because a fallacy has been committed in an argument, does not necessarily mean it is inherently false. It simply means that more work needs to be done to it.
"When you tell me we should be doing what other affluent countries are doing by providing universal health care, you are committing a bandwagon fallacy. Therefore, we should not support universal healthcare since the argument rests on fallacious reasoning."
It's pretty clear here that just because someone commits a fallacy in their argument, it does not necessarily make it false or wrong. It just means further work needs to be done if it is to be proved logically.