I think the question of the burden on the taxpayers purse is a strong second to the fundamental question of relevance within a modern, progressive society.
But for the purpose of the debate, I think the income that the royal family makes from its wealth and ownership also has to be taken into account.
For example, as the Duchy of Cornwall Charles, and now William, make circa £20m p/a from the 150,000 acres of land that accompanies the title. That's one small example, the royals own an astounding amount of land, even before other assets are accounted for.