@MisterChips No muddle, I've stated that above :) and of course they do benefit from 0 VAT when building new accommodation ( so there is the ability to claim it from construction).
I do note you went to lengths to prove your point regarding the 8-12k
I'll just do one thing, the average VAT for a school fee in the UK would be about 3200 ( the average of the primary school and secondary school averages)
Take that from your top figure of 12,000 and you have 8,800.
This is before you count in discounts from business rates, the VAT exempt status on several things, which would lower this below the 8,000 limit.
Of course, I'd also wager that the the private school figures are calculated across the total private school population, forgetting that 5% of students come from abroad with their parents living there. I'll use this figure and be generous because there are another 5% whose parents are foreign nationals and the students were born abroad who study at private school, the likelihood is that these students would not have access to state school places either.
This means that the estimated saving to the exchequer has to be lowered again. So, yes. I stand by my "wildly overestimated" just from the cash figures alone.
I'd also say that if we're doing a true cost/benefit analysis we should look at other costs and benefits too, but we'd need shadow pricing for that, and it would be too complicated for MN discussion.