Except you are leaving out my qualifier "automatically" and are not taking into account verbal logic employs semantics, unlike mathematical logic which does not.
The statement is qualifying that "southern" doesn't mean "polite", but rather it means automatically or implicitly polite. Sure other types of people can be polite, but they are not presumed to be polite by her statement.
If Schlitz = a beer, not all beer then being out of Schlitz does not make you out of beer.
But if Schlitz = good beer* versus other beers, then to make a statement "we are out of Schlitz" implies in verbal, not mathematical, logic, that you are out of good beer. Otherwise why bother to qualify what kind of beer you are out of? It becomes excessive verbiage, which is generally not a good thing (hence editing being a whole profession, equally important to writing).
Semantics do matter. They might not to you, and that's ok. But to me, her statement was ruder then it was polite. She qualified her manners in such a way as to come across, to me, as polarizing.
* All hypothetical of course, since I do not agree with the premise Schlitz = good beer