“The way I see it is, that once the sharp elbowed parents are in the state system, they’ll start to agitate the government to improve things , which will be better for all the students.”
You don’t mean “sharp elbowed” parents though, do you? You mean influential journalists and MPs, people with power to sway political opinion?
Sharp elbowed parents like me - we support our schools, fundraise, tell everyone how great our schools are, buy the head teacher a bottle of wine for Christmas, email the teachers praise and thanks and make sure our DCs do their homework, study for their GCSEs, don’t forget their kit and most importantly, if they are disciplined at school we support what the teachers have said! We understand that the more desirable the school - the better its future; the more funding, the less problematic families. The better results reflect the cohort primarily, as well as teaching. We want our kids to be happy, in a nice environment. Every child deserves that.
The journalists sending their kids to state schools is already happening and precisely the reason why this VAT policy is coming about anyway. Because they shout the loudest, comfortably sitting in the most sought after state schools. Which by the way are getting bigger and bigger, at the cost of the failing ones, full of problem children and disadvantaged families.
Here is a good example of a successful huge one, expanding even further:
https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1174393/ADA4215_Kingsdale_Foundation_School_Southwark_25_July_2023.pdf
The middle classes decide a school is good, Ofsted brands it with Outstanding, good teachers stay, good vibe, great results, happy days (mostly). Has zero effect on a neighbouring failing school. Nobody is going to improve standards for everyone. Why would they? It has nothing to do with private schools and all to do with poverty, deprivation, NHS funding, extra help when kids are struggling, benefits, social services, adequate housing and most importantly lack of drive and mental health issues etc.