I imagine they bring in more money through tourism than they cost.
Americans are crazy about the royal family.
They bring in more money from taxation than they cost. Far more - the cost is 15% of the
voluntary supertax they pay. It's defined that way.
The other costs are either standard costs of a head of state and diplomatic functions (e.g. while it would be funny to give President Trump some vouchers for a budget room at a Premier Inn, it wouldn't be good diplomacy) or standard costs of building maintainence. Of course, all that would remain a necessary cost if we deposed the royal family and stole all their stuff.
So what's actually being proposed is simply stealing everything from a rich family. But only one rich family. Not other rich families. Not usually, anyway. Some people are more consistent and advocate enforcing a wealth cap in general rather than targeting only one family.
Tourism is also a thing. Yes, tourists would also come to see a palace by itself. But not as many as would come to see a palace a monarch lives in. There are far prettier palaces elsewhere. Disney probably has some. Apple, Facebook or Google could build one in a couple of months and it would be easier for most Americans to get to. But vast numbers of tourists wouldn't travel to see Google Palace just because it was pretty. Even places like Taj Mahal and Versailles don't get tourists solely because they're pretty, and they're probably the prettiest buildings in the world.
As for the name, it's clearly a Cunning Plan to take over France. The British royal family technically has a legitimate claim to the throne of France(*) and giving a British prince a French name is "obviously" the beginnings of a new takeover bid.
* That bit isn't a joke. At one point, the King of England had the best claim to the throne of France and the French claimaint only got the throne due to bigger army diplomacy.