It may be a widely held view, but as an American who works in the education system, I can tell you that it's an untrue and oversimplified one, and one that unfairly paints an entire country of incredibly diverse people as uninterested and uninformed.
No kidding. The monarchy is such a huge tourist draw for the UK -- they need to keep up these palaces if they want them to continue to be destination places for people who want to experience living history.
Also, it'd be a huge shame if HM and The Duke had to start carrying one of those plastic see-through umbrellas indoors at all times!