... The Monarch also gets to choose which are "private" properties, and not part of the Crown Estate. ..
That isn't true
William the Conquerer introduced feudalism - which meant the monarch nominally owned all the land
The Civil War established that "the people", in the form of Parliament, were Sovereign over the Crown - which meant the end of feudalism
But the Monarch still owned a vast property estate - the balance that was struck was that the Monarch kept ownership but out of the revenue they had to pay for most government things (like the Civil Service and the countries defence for example)
George III changed that - rather than the Monarch having to fund government things he gave the revenue to the government, and they in turn gave the Monarch money to live on
The Crown Estate is neither the property of the government, or the property of the reigning Monarch - it is basically its own entity
The government gets all the revenue - but with an agreement to pay some of it to the Monarch
-- Completely separately to this, sometimes the King or Queen buys their own property
Sandringham was bought in 1862, Balmoral was bought in 1852 - they were just bought like anyone else would buy private property - nothing to do with the Crown Estate
The Monarch has zero control over the Crown Estate, they do not manage the Crown Estates and can't take things out or add to them.
Whether they should have tax concessions is another thing - that's very complex. Basically all very wealthy people should have tax concessions because it's very easy for them to avoid paying any tax's - giving them concessions is basically a way of tricking them into paying more tax than if you gave them nothing. No concesssions - they'll try and avoid it all; with concessions they'll pay less - but more than they would otherwise. And the actuall amount they pay is vastly more than everyone else so it's a pretty vital source of revenue.
