"I think that if you want to look at the effect, it is far better looking at the percentage of pupils privately educated, as opposed to schools closing or entry numbers without adjusting for demographic decline.
And, according to government statistics , the number of privately educated pupils is fairly steady at 6.3%."
That seems a fair point, in your first para, but is it overly simplistic? It won't capture eg, any pupils who are now educated abroad (either 'foreign' families who no longer send kids to board here, or UK families who decide to move abroad due to overall tax burden). Though I accept these will be a very small proportion of the 6.3%.
I believe more SEN independent schools have opened so the move of state pupils to those will mask some movement going the other way from mainstream indy to state. But again, it may not be a statistically significant amount.
More interesting are these government statistics, for the 25/26 academic year, released last month:
"The number of pupils in schools in England has decreased
There has been a decrease of 112,200 (1.2%) pupils across all school types from the previous year, with a total of over 8.9 million pupils. This includes all state-funded and independent schools."
contrasted with
"The number of pupils in independent schools has decreased by 3.8% to 560,300, declining for the second consecutive year."
So state schools have seen a reduction of just over 1% (stat of 1.2 above is for state+indy) while indies have seen a 3.8% reduction - well over three times that in state.
I'll admit that I'm not a statistician, and that I also struggle to reconcile this with the 6.3% 'holding steady' figure, though from a quick and unchecked google search it seems that this has fallen over the last two years from 6.5%.
Release home - Schools, pupils and their characteristics - Explore education statistics - GOV.UK