I don't think anyone gets rich from museums. The already rich get involved in museums.
Also, lots of popular entertainment gets no public support- rock music, movies.
I agree that some amount of public funding for stadiums is probably a good idea, I'm just not sure that "popular or worthwhile=needs support" holds up.
edit: To get a bit more nerdy, I think that the kinds of things that need public support are the kinds of things where the benefits to society or a region are widespread, but those benefits don't necessarily accrue to investors. Situations with positive externalities. Sports franchises might be one of those things.
It's just irritating when teams go to state and local governments for special favors year after year, but then when they're not happy all of a sudden they act like the team is their property in the same way your lawnmower is. At lot of the value of the team comes from civic support to begin with.