"they come and spend oodles in shopping areas as well as going to see the sights, tour buses are usually packed"
They do indeed, but you are surely not suggesting that this would change if there were no royal family... Not even the dumbest American tourist believes there's any chance of meeting the queen.
According to travelandleisure.com, of the 25 most visited attractions in Europe, Britain's first entry is The British Museum at number 13. No royal palaces feature at all.