Which tourist attraction brings more money in? A palace charging entry at £10-20 a time, or a mostly empty private house where people stand in the street for free?
Depends what other places they go to while they are on holiday here

And actually
some people do pay to visit parts of it. (As an aside, I don't think in purely financial terms, and I would suggest the pleasure people gain from visiting free attractions is important, although I realise that's not pertinent to a discussion of how much money is generated by tourism).
Do you really think there would be a net loss of income to Britain - as in, people, large scale, deciding not to come here, if the royals were shuffled away?
Yes, I think there would be a net loss of income, although I don't know how big (and I should stress, that's not my main reason for supporting the monarchy). I think their popularity and the emotional attachment that many people in different countries feel towards them is very self-evident from the crowds on their foreign visits, and the enthusiasm with which huge numbers of people follow news about them. Since the royal family is so closely associated with the UK, it would be very implausible for that not to translate into a desire in some people to visit the places that are associated with the Queen etc. - most obviously, London. (I realize though that it would be very hard to measure the effect precisely).
I'm going to hazard a guess that you personally don't feel much emotion towards the Royal Family, and perhaps therefore you don't really get the way many other people feel towards them, or the happiness that many people derive (however illogical it appears) from the existence of the monarchy and the respect/admiration they feel?