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With that being said, the current policy doesn't seem to be to scrap them entirely, but to remove their charity status.It does not make sense to scrap their charity status.
As you said, people who send their children to private schools are not necessarily wealthy. But in any case they do pay their taxes yet they do not use state schools' resources, which actually means more money per state school pupil. Why would these people have to pay extra taxes for not using state facilities that they already finance? (Because that's exactly what this would mean)
Private schools help the system. Hitting them would hurt everyone.
That being said, at this year party conference they did vote to scrap private schools altogether, to seize their assets, and tho redistribute these assets "democratically".
> they're doing so to push the class divide
It is an ideological class war. This is communist.