There's no "bigger picture" thinking behind this idea.
An increase on private school fees won't bother the super rich; they'll continue to pay and private schools (& potentially universities) will become educational facilities for the uber-rich only.
Those mid-high income families barely scraping by to put their kids into private will pull them out and spend the money on buying houses near the best state schools/grammar schools making it impossible for those on a lower income to afford to live within catchment.
State schools will become oversubscribed when families pull their kids from private and the catchment areas made smaller, so only those who can afford to live very close to the good schools will be able to get in.
This trickle down effect will continue. It will take years, but the class & educational divide will become even bigger.
All for a relatively meagre tax income which will be a drop in the ocean compared to all the state funding the government will need to stump up.
Instead of parents paying £20k per year per child to get their child educated (& it costing the tax man nothing) if that child is placed into state then the government will have to give £5-£6k per yr per child (I believe is the current figure) to state schools for that child, who previously cost them nothing. The tax would have raised £2k per year, but instead will cost them £5k-£6k; a £3k to £4k loss per yr per child.
How is that good business practice and good for the economy?