You're conflating private school usage and wealth, which do not go hand in hand.
In our area the wealthy do not tend to send kids to private school- they move house to the higher priced houses near the highly rated state primary schools so that their offspring win the postcode lottery to get in. For secondary education there is no need for them to pay for private school either- they simply move homes if necessary to the schools with smaller class sizes that are not near locations with problems like gangs, drugs, knife crime etc.
The rich don't need sports activities or facilities provided by private schools because they are rich enough to afford membership to one or more country clubs and sports clubs with these sports on offer. They can also easily afford various private music, drama, dance etc lessons - if their kids are interested- and don't need these provided at school. If the local state school has staffing issues and the offspring struggle to pass GCSEs and A levels- no problem, Mum and Dad are wealthy enough to buy in private tutoring: even a different tutor for each subject if needs be!
The wealthy also know that wealth is a far bigger predictor of success and career opportunity than academic grades- why stress yourself and your offspring to work hard and achieve high academic standards expected by most private schools when it's easier to coast along and achieve mediocre grades at a state school then waltz into a cushy job opening offered by Dad, or Grandddad, or Mum (either in their own firms or through similarly successful or wealthy contacts in their social network) etc?
Most of the people I know whose kids went to private school cut out foreign holidays, shopped at discount stores (Aldi, Lidl), made do with an old car, scrimped on home renovations/repairs just to afford private school fees and are poorer than a good number of families with kids in state school who acquire new phones regularly, always drive new cars, take foreign holidays 3 times a year, and undertak home refurbishments costing over £20K. Sure, tax the rich. But they're not always found in private schools and a very high proportion of parents with children in private schools are not rich.
Private school is just simply another thing to tax, because they feel they can, like VAT on shoes or winter coats.