But I might be wrong. https://www.bbc.com/news/explainers-57559653
So while they don't rely on tax money (outside of the huge amount of tax money spent on their police protection, transport in military aircraft, etc. etc.), they do rely on money which rightfully belongs to the people of the nation.
Today, I expect that's what would happen. But I bet the Windsor family could have kept a lot if they had given up monarchy a while back. Which puts the "rightfully belongs to the people" claim in question, if you believe a king can ever legitimately own anything.
And yes, I realize the irony as an American. I think on average Americans like the British royal family more than Britons do?
The British royal family has veto power over parliament. They use this power to rewrite laws. Lots of this is well documented, but poorly publicised.
So they are great until they aren’t, and then there is no mechanism to remove them.
Dictatorships are also good if you get a great leader.