No this really should not be how it works either in schools or for the nhs. Such systems destroy and degrade the central principle of meeting everyone's needs without favouritism.
In both, 100% of people are entitled to a reasonable acceptable level of service, free at point if use, and no-one but no-one gets a better grade of service within the state system on the basis of ability to pay. Resources are allocated on the basis of need, not giving the nicest slices to the wealthiest (though that can happen a bit with selection-by-house-price in catchment areas of desirable schools, which is why many such schools are opting for lottery based systens rather than catchment)
Firstly, running a school or a hosipital serving X000 people does not become a set amount of money cheaper if it only has to serve X000-1 people. No significant savings are made, so there isn't a pot of "savings" that could follow those going private.
Secondly - it opens the doors to a spectrum of different levels of service offered on the basis of wealth not need. School A remains free and takes only the poorest and most disadvantaged children who have no alternative. School B is just a bit nicer, you only have to pay £1200 per year - £100 per month - to get a slightly nicer education experience and the great thing is that your kids can avoid mixing with the no-hopers in school A. School B chargest a mere £3600 per year top up above state funding and you get a significantly better education experience than those stuck in school A and while its a stretch, a lot of families can manage that with a bit of additional belt-tightening in their budget for Christmas and summer holidays. And the overall outcome is that the most vulnerable children get left behind and have the worst education. The whole idea ought to be abhorrent and is abhorrent to anyone who isn't totally selfush about advancing their own interests above those of people who are weaker and in greater need.
All children have a right to an education. All children can get that education completely free from the state. It is right that parents are at liberty to choose a different option outside the state. No there is no justification at all for them to get a singke penny of state funding to facilitate that.