Nellodee, did you read the paper you are citing? 50% of top 1% income sending children to private school doesn't mean that 50% of children in private schools are from top 1% income families.
In fact, if you look in Table 2 of the paper you cite, (“Income, housing wealth, and private school access in Britain” by Henseke et al which is open access, so it is easy to check) you can see that average weekly income of families sending their children to private eduction in 2018 was £429 per adult after tax (assuming 2 adults and 2 children in the household -- I am a bit overestimating, as they report household income divided by the square root of number of people in the family and I assumed 2 adults and 2 children) which amounts to £22308 a year and a house value of 366K in 2018 prices. I honestly don't think it is top 1%. Neither, incidentally, does HMRC: they put this income in top 51% (see https://www.gov.uk/government/statistics/percentile-points-from-1-to-99-for-total-income-before-and-after-tax#full-publication-update-history).
Thus, the picture of an average parent with children in private school is much closer to the picture painted by your opponents: people on average salaries with average houses rather than super-rich oligarchs (note that the mean is always higher than the median, so in reality more than 50% of parents with children in private schools have less wealth less than that). It is worth noting that another paper by this author (Anders, J., F. Green, M. Henderson, and G. Henseke. 2020. “Determinants of Private School Participation: All About the Money?” British Educational Research Journal. ) concludes that the main determinant of sending children to private school is value attached to the education by parents, not the wealth. So VAT will be a tax on aspiration, not on wealth.
Incidentally, the paper that you cited also debunks major assumption in IFS report (https://ifs.org.uk/publications/tax-private-school-fees-and-state-school-spending#:~:text=Key%20findings,%2Dday%20and%20capital%20spending).). Clearly para 2 doesn't make sense: in the cited period the salaries of people were growing by 2% and the house value by 8% a year (same Table 2 in the cited paper)-- this is why families could accommodate fee increases of 4% a year. Sudden increase of fee by 20% is totally different and the assumption that percent of the children in private education will still stay in 6-7% range is preposterous (especially if you take into account that families in 2018 were already spending 67% of their income on education on average according to the paper you cited). I don't think families can afford to pay 80%+ of their income on education, so we are looking at 4-5% of children in private education after VAT is imposed.
With this kind of numbers simple calculations show that the total income from imposing VAT (assuming average spend per school child of £8K by state, average private fee of £15200 and effective VAT of 15% as in IFS report) is negative -- it will cost the tax payer due to people moving from private to state. Not the effect intended.
Note that IFS assumption that families moving children to state will still pay VAT on other goods they will purchase instead (para 4) is unrealistic at best: people would, most likely, invest in private tutors/extracurricular (no VAT) and put money in savings optimising for taxes (still no VAT).