I suspect private schools is one of many issues which will prove popular with everyday members of the public, once the election kicks in and the media is compelled to report what Labour is actually proposing.
And no, it has nothing to do with envy.
Ensuring that rich and powerful people have a stake in the success of public institutions is the only way to prevent them from being trashed by oligarchs and their pet ideologues.
There is no chance on earth that Michael Gove and Dominic Cummings would have been permitted to unleash their hare-brained pub-napkin experiments on the schools if David Cameron's kids were to be subjected to them (or of schools forced to close on Fridays from lack of funds, for that matter).
Just as there is also no chance the NHS would have to endure idiotic PFI impositions like these:
https://inews.co.uk/news/health/nhs-hospitals-pfi-bill-ippr-report-toxic-legacy-556120 if the lives of the people whose taxes the Tories and Lib Dems cut depended on it, like the rest of us.