It's no more "classy" than London or Paris ¡ust because Mozart lived there for a couple years. City names like this really grate on me if the person using it has never been or has no particular relation to the city; I take Vienna sort of personally because I've been obsessed with the idea of going since I was 15 and am hopefully going there to study this spring.
Imagine for a minute that the city/town you live in is really famous, and that it was a city name before anything else, and that it happens to have trendy sounding syllables; imagine someone on BTN posting about it. Would you feel like they were cheapening it? Would you feel weird at all? If not I guess go for it. I would be irritated though, and I'd be irritated by meeting a Vienna. moreso than London or Paris even, as I don't have a personal connection with those cities