I'll bet that the majority, nay, the vast majority, of those millions of people will do their best to "game the system" when it comes to their children's education. I don't blame them. The game is set up in such a way that this is the best strategy for them as individuals. I use the term "game" in the academic economic sense because it is not a joking matter.
If you want to fix the system, start by funding public (not in the Eton sense) schools properly, so that all schools for the public can achieve good results, not just schools which manage to concentrate well-to-do, supportive families. You do that by raising general taxation by a sensible amount, by raising the rate of income tax on the highest earners, and by raising the tax rate on dividends and capital gains, i.e., unearned income. Playing around with VAT on private education and hoping that the derisory amount raised and that the families driven out of the private sector will take on the task of improving the entire educational system is akin to trying to bail out the Titanic with a teaspoon.
You cannot have good schools for all children on the current budget. And the £1.7bn, even if it all materializes, will not make the difference. When the state doesn't provide enough lifeboats for everyone, you can't blame people for doing whatever they can to grab places on the ones available. Especially when the seats are for their children.