I work a lot for public schools as an electrician, so am aware of the charity status, however due to this they are unable to claim VAT back, which pretty much balances out. One local bursar tells me they would welcome lifting the charity status as it would be beneficial to be able to do VAT and they would be better off.
I was at private school from 4 to 18, a mid range prep school and a mid range senior. I have never encountered any helping out amongst the old boys that you wouldn't find in the local comp. In fact I reckon the lads there have come off better, i'll give you an example, as boarders we came from all over, I have one old mate who now works for Rolls Royce in Berlin as an engineer, another is a GP in cornwall, another a general builder 200 miles away and one a barrister in London. we all do totally different jobs, plough different furrows, we catch up at reunions every few years or a coffee or beer if we are in the area, but largely it's a distant friendship.
Now my mate Chris, another spark who went to the comp down the road, where it seems most of his year group went in the trades knows all an everyone, can blag favours left right and centre, one of his mates from school is a GP too who returned to town after uni, and he seems to jump the waiting list at the local practice - I certainly don't think my education did me any favours in the favours department. All the elecution and deportment classes left us all with cut glass RP accents, mine does end up with me getting the mickey taken on site sometimes!
However, consider that every parent who pays for private, and many like mine scrimp and save for it, has the absolute right to send their little ones to a state run school at a cost of £5500 per year to the state (the standard senior school capitation), in effect by sending little Johnnie private they save the treasury the expense of his education.
Problem with Corbyn's ideas is they are not costed out, Labour never cost out their plans - Also he is all ready to scrap private schools, what about private healthcare? most surgeons spend half the week down the private hospital while NHS patients wait, if he is against private education, why not private health - I suggest it is because he has no need of a private school, but might just need a Harley Street quack sometime in the not too distant future.