It's not really "simply because the state also provides an alternative".
It's true that the tax-payer-funded alternative does make taxing education less economically sensible than taxing other things - because if the person then chooses not to buy education then the state is on the hook for £8k per child per year. Whereas if I decide not to buy chocolate because it's got too expensive for me, the state is no worse off.
But really, the argument is the other way round. The state provides an alternative because educating children is so important for society as a whole.
And giving a better education to children is also better for society as a whole. That's why there's always pressure for more education funding.
You might argue that private education isn't better than state education... but actually that weakens your argument. Why would you think that parents should be stiffed for £thousands for choosing to pay for a worse education themselves, when that saves the state £8k per year?
Then you get to the consequences. It's pretty evident from both economics and history that taxing something reduces how much is bought. That's why we tax cigarettes and alcohol.
So, if you follow those steps, you can see that taxing the 'better' education -> less of 'the better' education -> worse for society.