*"but if the numbers moving are large at a local level,"
They just won't be. There won't be large local level moves because private schools have far wider catchments than states, this means that even a relatively large number of students moving will come at a lower cost per head.*
Why do you think catchment size matters, except as a proxy for comparative numbers of children in private vs state?
What matters is the number switching (either immediately or just by not choosing private over the next few years), and whether there's existing space for that increased number within commutable distance.
So the concentration of private school uptake in a local area matters. And that isn't evenly distributed in the UK.
As I mentioned up thread, Edinburgh has 30% of children in private schools and Bristol isn't far off.
Do you really think that 30% of Edinburgh families are wealthy enough not to notice an extra £7k coming out of their net income each year?
If just 10% of Edinburgh families who would have chosen private decide to choose state instead, that will increase the number of kids needing a state school place in Edinburgh by 5% over the next 7 years.
There are currently 80 primary schools and 21 secondary schools in Edinburgh. Many of them don't have spaces, because this concentration of private school uptake has been in place for long enough that the council has factored it into their school provision.
So that's probably 1 extra secondary and 3-4 extra primary schools needed in Edinburgh (for 10% migration over the next 7 years)
Do you think that even 27% of Edinburgh families won't notice an extra £7k coming out of their net income each year?...