That's exactly it. This policy is being claimed to raise £1.5bn or whatever it is today, based on extremely optimistic assumptions. It only takes a small behavioural change (kids go state, parents reduce hours, a few school closures) to take some £hundreds of millions off the total. A less optimistic outcome sees losses>gains.
We should have a complete and honest appraisal, and it should reflect opportunity cost. The opportunity cost, of course, is "how we could raise money in better ways". Taking £1.5bn more in income tax, for example, whether you pitch it at the top decile, third, or half will
- raise money with much greater certainty
- require no legislation - and thus no avoidance/enforcement challenge
- not disrupt anyone's education
- ensure "the rich" that get great state schools today while buying VAT-free tutoring also contribute to improving state schools
- better reflect the social benefit of all forms of education rather than upholding the interests of state education only
Alternatively we could be like Finland and say EVERYONE needs to pay more taxes in return for better public services. But I don't hear anyone making that case.
PS: It's not a "get out" to say people will just move house. Moving house is a waste of resources and a huge deadweight cost to society and results in displacement of another child from whatever is the "preferred" state setting.