The royals do earn their own wealth even if they stopped being royal they would still be the wealthiest family in the world. The cost on the tax payer is covered by the queen and her income tax
Totally incorrect!
The Sovereign Grant Annual Report states that the Sovereign Grant was £37.9 million for 2014-15. She doesn't pay tax on this.
She doesn't pay tax on benefits in kind, like security and transport, etc, provided by the police and armed services. Neither does she pay tax on gifts. If being Queen were a 'proper job', all these would be taxable.
The royal family are exempt from income tax, so what they pay is a voluntary contribution. The details of the payments are private. The Queen pays a sum equivalent to income tax on her private income and income from the Privy Purse (which includes the Duchy of Lancaster) that is not used for official purposes. Anything she can claim was spent in pursuit of her duties is excluded.
They are far from the wealthiest family in the world. There are entrepreneur families like the Sassoons and Waltons; loads of other royal families from Thailand to Brunei via Liechtenstein and Monaco; various banking families and property owners. They're not even the wealthiest family in Britain.
The Queen's private wealth comes from investing on assets that were not taken over by the Crown Estate. Thus, it was not earned and you could say the same about most of our aristocracy. Admittedly, it would be hard to sort out how much of what they own personally 'rightfully' belongs to the people and how much to them personally. The Crown Estate was supposed to draw a line between them, which I suppose it does somewhat.