I would have taken mine at that age, and did take them to a similar place. It was bloody hard work, but not impossible. Lots of preparation in advance - talk beforehand about what would happen when we got there, the fact that they would have to talk very, very quietly as they did when we were in the library, etc, etc, etc. It helped that we only ever had meals at home sitting at the table, and I also took them to a lot of places when they were little which involved them having to behave in a way that didn't irritate anyone else. However, it also requires heavy parental involvement - mine were little pre-screens, so we always took books, colouring, quiet card games, noughts and crosses, hangman, I-spy, etc, etc, etc.
So it's possible, but requires preparation and hard work.
And despite all this superb parenting, there is now only one of mine (aged 16-19) whom I could reliably take to the Ritz now. 