That's because you believe (I presume) that ending the VAT exemption will cause parents to stop using private schools. It won't. It will only have the effect of disrupting the education of some children at private schools whose parents are not wealthy but have made great sacrifice for them to go there, or of making it harder for those parents to afford the fees. It will put additional pressure on the state sector and unfairly penalise children who need smaller, more focused settings as that provision is not available in the state sector.
Private schools are effectively now acting as paid special schools for a significant cohort of children. There is a significant cohort of children at my daughters school who have ASD, Mental health issues such as anxiety, anorexia, physical health issues such as Type 1 diabetes etc. Sometimes it is the only choice other than home schooling - which may not be a possibility if both parents have to work.
It is these children that increased costs will affect. Not the progeny of hedge fund managers and the super-rich.
I will take a random example of a small private school not too far away from where I live. It is not a world-renowned school - we aren't talking Eton here - it is probably a good example of an average private school.
Fees for yrs 7-13 are £8950 per term, or £26,850 per year. That excludes registration costs, one off fees, clubs, activities and many other sundries listed on their site as extra cost items.
Assuming tax at 40% then that would be £44,750 gross income required just to pay the fees. Even at 20% that is £33,562. Median working age household income in the UK in 2022 was £35,000 before benefits and tax.
Private schools are not the sole domain of the super-rich, but you aren't going to be able to afford going there if you are an average household for sure. This isn't a case of "making sacrifices" - scrapping the family holiday is very unlikely to even cover half of one terms fees. This is having a significant household income in the first place. Bear in mind a household income of £81k puts you in the top 5%, and after taxation, the modest fee paying school I used as an example above would be costing around 50% of the take-home pay before the schooling extras.
The point around special needs support is a good one. State funding has been woefully lacking, but I would argue from the above example that a parent of a child needing this care would not only need to be willing to sacrifice in order to pay the fees but be in a very high household income in the first place.
20% VAT on private schools in order to properly fund SEND and more general education in the state system seems a reasonable way to try to increase access to these provisions that is not reliant on the parents happening to be very high wage earners or independently wealthy.