The Crown Estate is where the sovereign grant comes from, and it's weird. It's "owned" by the current monarch, but only in their role as head of state (the Crown). Realistically it belongs to the country and to the public, and were we to abolish monarchy, would not become the private property of the Windsor family.
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.
> were we to abolish monarchy, would not become the private property of the Windsor family.
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.
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.