Yes except that private schools won't be abolished; that would be a step too far for even Corbyn and his Momentum cronies. The threat is the removal of charitable status which exempts them from charging VAT on their fees. Even if this happens, it is unlikely to affect Christ's Hospital as, despite the number of full fee-payers, the majority pay far less, whether on a bursary or a scholarship. For those schools unable to demonstrate their charitable nature, which may well be fewer than people like Mr Redshaw like to think, fees will obviously increase.
Some, though by no means all, of these schools will inevitably go to the wall. Independent schools are a remarkably marginal business in many cases even with charitable status. The rest will continue just with only the children of somewhat richer parents than at present. Most people I know who send their children to independent schools make significant sacrifices to do so, but the imposition of VAT could well be the straw that breaks the camel's back.
The outcome: fewer independent schools undoubtedly but those that remain would be taking children of significantly wealthier parents, thereby further increasing the gap between rich and poor and limiting the opportunities for those kids involved. Of course a percentage of the teachers from the schools that close will return to the state system, some will not. Possibly some of the schools that close will transfer to the state system; there are examples of this, for example, in Bristol: Colston Girls' School, and St Ursula's for a start. A substantial number of school buildings and a percentage of teachers, however, will undoubtedly be lost to education in the round. This will increase the stress on the state schools where the displaced pupils and, more significantly, the next generation who might have gone to an independent school but now won't, end up.
There is one thing on pretty much everyone involved in education agree: very large class sizes are a bad thing. They adversely affect the education of the pupils and increase the stress and workload on the teachers. This is just one of the inevitable outcomes of any decision to remove charitable status from independent schools. Another is that an influx of previously privately educated children to any school is likely to overload the higher sets. If you want a decent grade (6 - 9) at GCSE in any of these schools then you definitely need to be in the higher sets. Though brighter pupils in mixed ability sets tend to raise the standard of the lower performing it is invariably to the detriment of the brighter. The effect will thus cascade down the school; pupils who fall out of the higher sets and into the lower will do less well as a consequence of any pupils joining their school from independent schools.
Now if you are an independently educated Trotskyite like Mr Corbyn (or Mr Redshaw for that matter), none of this is important. Application of Socialist dogma is all that matters and this is a policy that looks good on paper. Why should the better off get any benefits from the State; their job is to pay the taxes not benefit from them? Unfortunately the reality is likely to be somewhat different. The stress on our already creaking state education system will be increased in a number of areas and the overall quality will diminish. I have no doubt that a Corbyn government will increase education funding but recent history seems to suggest that most of this will end up in new buildings rather than more teachers and other staff.
At the end of the day,discussions on education tend to come down to articles of faith. I believe in a degree of parental choice and that independent schools are, in general a good, rather than a bad, thing; many others do not. It is not, however, quite such a simple argument as it might first appear.