@MrChips But the accommodation measure IS used, is doesn't matter if it little is, it still comes at a cost to the treasury.
Ok, taking your Average VAT + 0.1k on business rates, means that at the estimate of 8k saved, comes down to 5. 2k and at the top of the estimates comes down to 8. 2.k BUT...
State schools also don't have to provide transport for children out of catchment, its only provided if you go to they go to the nearest suitable school and live more than 2 miles away for under 8s and 3 miles away for over 8s. The fact is that this extra cost applied to all private school children is erroneous as most will live within 3 miles of a suitable school. Even then in London all children under 18 are given some type of free or discounted travel, which given the massively disproportionate number of private school students in London, would further reduce your erroneous per head calculation. This also means they currently do receive some sort of state funding for travel, something you didn't add into costs to the exchequer for private schooling.
The thing that is laughable is that to get to your 12k figure you added costs which aren't divided per pupil and would be spent anyway with thousands more or less in school), so come at 0 opportunity cost and wouldn't be included in a per head calculation in how much the state saves from private schooling. Nor have you considered that private schools may also benefit in some way from R and D spending by the DofE. So these are easy to dismiss from any per head calculation used to say that the state has saved from not having to educate these students.
Your simplification to get to 12k also assumes that schools would need to spend 12k to accommodate each extra child in order to provide for them, when if you spread out the entire private school population across the state school population the figure would be something like 1.1 child per school, so far closer to the figure of 8k at the lowest.
In actuality no school, even if ALL students left private education, would be adding 20 students to a year group and therefore have to provide extra spacing, an extra 20 students divided even across 1 year group wouldn't come to the 8k estimate on saving.
And as said you didn't take off the cost of the VAT tax break business rate reduction from the figures you used in order to come up with the 12k figure, and made some wild and inaccurate assumptions on the costs to reach it
Lets make it even more simple.
Take the 2023-24 figure of 7460 and minus just the cost of the tax and business rates per child from the private sector being 2.8 k, this gives us 4660. All in all, So yes, the state is maybe saved about 2.9bn in having private schooling, rather than the maximum of 7.8 billion (and minimum 5,2bn) you suggest, so yes they are wildly over estimated savings. The even better thing is that so few children will leave the private sector ( all the think tanks are under predicting how inelastic PED for schooling is) that this saving will be added to a 1.7 billion, or more, increase in spending for schools.
Double win.
Adding on all the tax breaks for state school students, dividing capital expenditure and R and D from DofE onto the figure, whilst not taking the cost of tax breaks from private schools into account on your "saved" figure massively undermines your point and thoroughly shows the mental gymnastics you are prepared to go through in order to try to prove.
It is fun showing that though, but maybe you didn't understand that.