It would be hard to know what numbers to run, since we don't have a non-monarchical Britain to use as a control, although of course many of the most visited estates are managed by the National Trust, and don't seem noticeably less popular for the absence of resident aristos. I frankly doubt that the country's tourist income would be hurt at all, but even if it were, people making that kind of argument have to my mind already sold the pass. If the best reason you can give for keeping the head of state as a hereditary position is that it might entice foreign tourists to buy souvenirs, then you obviously don't have any serious arguments.