"I'm not bothered about the cost to all of us so much as the principle of giving extremely wealthy people even more money."

OK, so suppose we had an elected president. That role would still need to be funded. They would need accommodation, staff, travel expenses, the cost of official functions, etc etc. That cost would fall on the state, regardless of the incumbent's personal wealth, or lack of. That is what the Sovereign Grant covers - the costs of carrying out the role of head of state. It is not for the Queen's personal expenditure, that comes from her private resources.

Of course these costs could possibly be reduced, but that has nothing to do with monarchy v republic. Some monarchies run very modestly, some presidencies cost far more than our Royal Family. It all depends on what the nation wants and expects from its head of state. The majority of British people seem to enjoy a bit of spectacle and flummery.