Of course it costs the taxpayer. The government could simply stop handing over millions of pounds a year to these people if it wanted to. But it doesn't. This decision costs the taxpayer.
Taking the logic further - all privately owned property and revenue "costs the taxpayer", relative to the public owning it instead.
The UK treasury is formally Her Majesty's Treasury, and the sovereign grant paid to her is actually a small portion of the money she puts in through the Crown Estate. So she is a net donor to the treasury, although she gets a major out in that the Crown Estate is not subject to inheritance tax.
Of course it costs the taxpayer. The government could simply stop handing over millions of pounds a year to these people if it wanted to. But it doesn't. This decision costs the taxpayer.
Taking the logic further - all privately owned property and revenue "costs the taxpayer", relative to the public owning it instead.