‘Prove’ was historically used as a synonym for ‘test’, which gives the phrase quite a different meaning. Like how ‘result’ is now sometimes used to mean ‘positive outcome’, as in a football fan saying ‘we got a result’

The expression is referring to an implicit or unstated rule. Defining it is hard but people know when it has been broken. Hooters is an exception, the rule is, don't be like Hooters.

It’s a folkism, but consider this: If a rule doesn’t have any exceptions, is it really a rule? If a rule doesn’t exist how could there be any exceptions?

This phrase only made sense when it was explained to me that it’d be better phrased as “the exception that proves the existence of the rule”.

(if there were no rule, there wouldn’t be any exceptions to it, or nothing would seem exceptional with respect to the (previously unstated) rule)

In my experience a bar (not a pub/居酒屋, a バー) will frequently be a place that always has female staff working, where those staff will be wearing makeup and at least somewhat attractive clothes and expected to converse with customers. Not always - there are definitely bars focused on music or some hobby or particular kinds of drinks - but often enough that it's what I'd expect if you just said bar.

Think the takeaway was more about the "seediness" aspect of Hooters do to it's being pretty exceptional/unusual in American dining culture.

Hooters is a pretty unique restaurant experience in the US and is therefore considered different/further from the norm and frankly by many seedy. If there were more places like Hooters in the US then this would probably not be true.

The comment was trying to explain that in Japan you have a lot of places that would be analogous to Hooters in the US...so it's not exceptional/not seedy. Maybe not quite the "norm" but common enough to not be really something that gets noticed or have a connotation like "seedy".