I completely agree that in the short-term, most families will tighten their belts and swallow the rise in fees rather than disrupt their children's schooling. I also agree that in general, this will therefore not result in councils being flooded with applications, particularly combined with demographic changes. Additionally, I agree that we're not likely to see that much effect on house prices, because moving house is an expensive and slow process.
But the devil is in the detail:
What do we mean by "most"? Estimates vary wildly and seem to depend entirely on the political bias of the estimator. The tipping point where the measure ceases to generate any extra money also seems to depend entirely on political bias.
What do we mean by "tightening belts"? In the specific context of the wealthy private school families being targeted by this measure, the belt-tightening would reduce their VATable expenditure on non-essentials. I guess that's VAT we don't want to collect.
What do we mean by "in the short-term"? I'd be interested to know if anybody was predicting a particular flurry of movement in the coming weeks! Apart from urgent situations, doesn't it make sense for children to move schools at natural break points? So you'd see kids not moving to private at (e.g.) Yr3, Yr7, Yr9. And you'd see kids moving to state at Yr7, Yr12. This may mean parents having to wait a year or two.
What do we mean by "flooded with applications"? The distribution of school-age children and schools is not uniform throughout the country, or even within local areas. It's not exactly helpful to tell people in Yorkshire that schools are empty in parts of London. Where I live, school places are like gold dust and local children at normal entry points have been offered places practically on the other side of the city.
What do we mean by "house prices"? Again, it's all about very local situations as above. The previous city we lived in, there was a leafy naice area with 2 schools within a couple of miles of each other. One rated outstanding, the other not. It was amazing to look at house price difference within 1 street of each other, based entirely on the priority area for the outstanding school. Certainly it was the kind of house price differential to exclude all but the most wealthy. The city we live in now has exactly the same situation.
Bottom line to me is that the OP is correct - we do not know whether this measure will actually generate the projected VAT bonanza and how much the net financial/cost benefit will be. Even the most optimistic projections seem to put the figure at around 1% of the education budget. Not to mention that obviously this bonanza isn't going to be ring-fenced for showering on schools.
It would be more honest and admirable to say that this is an ideological measure for signposting what is morally right and wrong. Meanwhile, the general principle that "sinful groups" (private school kids, foreign kids) can be excluded from perceived "desirable" resources (grammar schools) has been firmly established in the public mind.