Don’t be ridiculous, there’s no “children’s education tax” - you make yourself look silly even calling it that.
It’s the removal of an unjustified exemption from a luxury service - free education exists for everyone, so there’s no need for anyone to purchase this.
Therefore it’s a luxury good/ service and should be subject to VAT.
It’s also something that is to the detriment of the country as a whole - removing some children from the same system as others and so the family’s participation and “stake” in the system - and so no reason for the government to encourage it was a tax break. If anything, it could have a higher rate of VAT to discourage the behaviour.
It’s ironic that right-leaning people are so against this, as they’re normally very in favour of VAT as a tax, with it being a regressive tax. The only world where having this VAT removed would be a good idea is where VAT as a whole was abolished, and everything rolled up into a higher income tax with progressive tax bands.
And no, you’re not paying twice, or subsidising other people’s children, any more than someone who never drives is subsidising the roads, some one who dies young is subsidising the pensions, etc.
If anything your friend could reevaluate her friendship with you but it sounds as though she’s risen above it.