Surrey has 40,000 children in independent schools (across both primary and secondary), 92,500 children in state primary and 65,500 in state secondary.
https://www.surreyi.gov.uk/dataset/2y3j8/number-of-schools-and-pupils-by-type-of-school
Whilst they don't break down independent school numbers between primary and secondary, we know that dropping birth rates mean that there are fewer primary school children than secondary. So I'm going to estimate that the 40,000 independent students breaks down to at most 10,000 primary and 30,000 secondary - since that would even out the total numbers of students across the 2 sectors between primary and secondary (if anything, more of the private school students are in secondary to reflect the falling birth rates).
Which means that at least one third of secondary students in Surrey are in private schools. 30,000 private school secondary students compared to 60,000 state school secondary students.
So yes, that would definitely cause a huge headache for the council if a significant proportion moved. Or if a significantly higher proportion of year 6 children chose state in Year 7 than have in previous years.
Edinburgh and Bristol are similar. Those are cities not counties though, and some posters have suggested that more private school students commute into those cities than state school students. That obviously doesn't apply for a whole huge county like Surrey.
It goes without saying that parents should only request a state school place if they think they may take it up. Doing it to cause trouble would be a really shitty thing to do. I still haven't seen anything to make me believe that they are: it really does look like rumour and urban myth. Possibly put out there by people (not private school parents) trying to stoke up tensions.