Apart from the immense wealth, privilege..
But that's the point. You and I have immense wealth and privilege compared with people in the third world. The other stuff is by the bye. We weren't elected either! Who cares how people got there... the point is there is huge disparity in the way wealth is distributed throughout the world. All well and good looking up and moaning. Take a look down and realise, relatively speaking, that you are obscenely rich yourself.
Tourists come to Britain for its rich history, beautiful scenery, museums etc, not for the remote chance of catching a glimpse of a royal.
Well, I don't accept your premise, (I saw an interview carried out with people who had travelled from Brazil, all because Richard III's body had been discovered), but even so, you miss the point.
Our thousand year old monarchy, together with tales of monarchs such as Henry VI, Elizabeth I, Victoria, Edward VIII, is part of our national identity. People don't come here to see a monarch of course... but the monarchy and associated pagentry is something that is distinctly British.
If we disbanded it would we see the effect tomorrow? Of course not. Next year? No. In another few hundred years? I think possibly we would.
Look at, for example, Royal wedding viewing figures..according to multiple sources approx 2 billion people watched the last one. People are interested.
I for one am happy to have an apolitical head of state. And in my view, as soon as you are elected, politics rears its head.