As far as I know they generate more money in tourism etc than they cost to fund. Also they have no, or at least exercise no powers of authority that I'm aware of so let them have at it.
That's true, but it's even clearer than that:
A large part of income from the monarch's own assets (which they inherited, so stealing those assets from them would entail removing the legal right to inherit) is given (voluntarily, by the monarch's choice) to the state. The state gives the monarch 15% of it back to cover the costs of being head of state. So the state profits a great deal from the monarchy even if tourism is ignored. Doing that would be deceitful, since tourism brings a lot of money into the UK and a large part of that is from tradition and culture, especially the monarchy. Any even vaguely wealthy country or major company could build a palace more impressive than any of the royal palaces, but it wouldn't do much for tourism. It's not about the building, it's about the heritage and the fact that a monarch makes it a "real" palace.
In any case, the monarchy costs little or nothing to fund. Without a monarchy, we'd still need a head of state and all the infrastructure behind them so we'd be spending about as much anyway. Maybe more, since a President would expect to be paid.
Even if they did cost money that wouldn't otherwise be spent (they don't) and they didn't increase tourism (they do) and they didn't give the state almost 7 times as much money as the state gives them (they do), it would still be worth the minimal cost to maintain part of our cultural heritage.
EDIT: The last time a monarch of the UK formally exercised royal power was in 1708, although it was exercised informally to force voting reform through in 1832 which ended the rotten buroughs, laid the foundation for universal suffrage, secret ballot and generally the whole of our modern democratic system.