From what I can tell, there's no voucher system or equivalent that allows the existence of competition between those schools, which is exactly why the private schools would (according to the proposal) be better.

Why are we trying to pretend that meaningful competition is possible in this sphere, or that it would produce a better result for the student?

Pretending is fun. Why is meaningful competition not possible in this sphere?

The same reason that meaningful competition is usually difficult (when it is difficult)--information asymmetry.

Experts don't really know what makes one school better than another. Is it performance on standardized test scores? Improvements on standardized test scores? College acceptance rate?

Even when you do decided on a metric, one of the biggest impacts on educational outcome is parental involvement. Is school A really more effective than school B, or just more effective at attracting involved parents?

If experts have a hard time judging the effectiveness of a school, most parents don't have a chance.

For an example of what this looks like, take a look at for-profit colleges. Through advertising and information asymmetry they attract students who will rack up tens of thousands in student loans for mostly worthless degrees. The most successful for-profit colleges aren't the ones that are the best at educating students, they're the ones that are the best at marketing and sales.

Late to reply. Funding is based on the number of children in the school - so if the school can attract more people it will get more money. Obviously this isn't the same as 'your tax' funding your children - but hey we at least play at being socialist. In major cities it is not uncommon to have a choice of schools open to you (I previously lived within 500m of 4 primiary schools) and so the competition to get both the right number of pupils to fund the school, all the while getting the 'best' pupils exists. It just isn't providing the results you are hoping - I'd say as it 'seems' that the funding ends up being used to pay the heads and other senior staff very well: https://www.telegraph.co.uk/education/0/academies-pros-cons/