Two things can be true at the same time.
Tax rises are BAU and arguably needed.
Some modifications made to the tax regime are done so for ideological or ethical reasons and it is a well known mechanism to influence public behaviour. This has been true for alcohol and tobacco products for example. The state decides that these are bad things, cause harm to society so specifically targets them with tax increases and special taxes. Private education is viewed by this government as a negative thing. Adding VAT isn't a politically neutral move nor is it the most obvious way that a government would look to raise money.
The toxic psyche comment is valid because unlike alcohol and tobacco it is harder to demonstrate that the tax will actually raise any meaningful money in the long term or that private schools actively harm society, especially in the context of a state system where inequality and failing to meet needs is rife. Why tackle private schools specifically when there are bigger fish to fry? The move is absolutely laced with envy and toxicity. Screw the people being failed as long as nobody gets more privilege than me.
I do think looking to the law to prove the point won't work. The government charges VAT on energy bills whilst claiming they want to keep bills down. The courts can't change bad policy just because it's bad.