Understandably, there are many who view private schools as a "luxury", rather than a piece of a healthy educational ecosystem (where variety and choice is essential). I suspect many do not have children currently in school, or the issues don't affect them personally so of course they don't care.
But there are other pieces of the educational puzzle: nurseries for example. There aren't enough state nurseries, so the vast majority of parents have to use private ones. Any extra tax on the sector would be devastating, similar to private schools. Private nurseries are often the only way two income families can exist. The fees for these are also often the same as private schools as they face the exact same pressures: rent, staff costs, utilities, tax etc. Therefore the inflationary pressures on nurseries are the same as that on private school fees, they have grown at about the same rate over the past ten years. People saying school fees have risen faster than inflation simply need to look to nurseries to see the exact same trend and the same line on a graph.
Private nurseries are an essential piece of the childcare provision, but similar to people's posts above, "they are a luxury and should be taxed" as state provision does exist. If the government decided tomorrow that they wanted a behaviour change and only state nurseries were encouraged in the spirit of fairness - and they did this by introducing a 20% tax on fees for private nurseries, what would those same people say? The outcome would be the same: private nurseries would close as they would lose customers and would no longer be able to balance the books. Parents would then need to scrabble for places at the nearest state nursery, but have no choice in allocation - what they are given could be too far from home or work, or with very limited hours. The benefit would be that it's free, but at what cost? What if the provision doesn't suit your baby at all, or there are severe staffing issues, lack of funding, overcrowding, or a child is hitting your baby. But there's no other nursery you can move them to because there is no choice and what's left of private nurseries are only for the very very rich. It's not so different to the schools landscape. Or university provision.
I would much rather the cost of alternative provision be brought DOWN rather that go up via regressive taxation as then lots more people have more choices over how their children are looked after and educated at all stages of babyhood and childhood.