@lemontt makes a very good point.
If the land was given to the State, the Government (whoever it is) would very likely sell it off to the highest bidder, who would charge a commercial rent wherever possible.
Many years ago I was involved (at a very low level) in the sale of hundreds of jobcentre buildings to private firms. Most of the jobcentres stayed put, but the rent has increased regularly and significantly. It was done as a way to reduce the costs to Government (liabilities + structural wear & tear) and inject £££s into the Treasury. Most are probably owned by pension companies now, with funds held overseas as standard, which could mean (I'm not a tax adviser) no corporation tax is paid either.
It is unlikely any institution (especially the State, given it's past record) will permit a peppercorn rent. They are more likely to sell off the buildings/land to release the equity. They're in this for profit, not the common good.
The Royals are no worse than anyone else here. At least (according to a PP) they pay some income tax here. Many in the same position would arrange their finances so that they pay none.
Clarifying my previous point on Government finances: if it sells off a chunk of buildings, they are no longer creating regular (eg maintenance) costs on the balance sheet, and giving a healthy cash injection. The occupier, for example, a Jobcentre/clinic/whatever will have to pay rent to the new owner (just as it did to the previous owner) and, very likely, all of the maintenance costs. If this is a service funded by a Government department, the expectation is that they will have to cover it. If they don't have enough money, it is usually claimed that the tenant isn't spending their budget efficiently. Not that the Government offloaded it due to the exact same costs.
Sigh. It's just another way of moving all the money around, and ensuring owners get most of the risk, and tenants get most of the bills. Nothing new, it's been going on for decades.