I'm afraid I think Tristam Hunt is speaking nonsense. I have close family and close friends in both state and private schools. What he's suggesting is impractical and likely to be ineffective. Many families using private schools are actually the typical "state school family" and vice versa - depending on where you live and sometimes social circumstances.
Some friends have the resources financially to be using private schools but have bought expensive homes near the state schools that are continually the best rated so in effect may as well be in the "privileged" private sector that gets great results, etc. On top of that their clubs and after school activities are much cheaper so they can spend their money on other things that boost their children's school results if they wish to - one has used the money saved up to pay for tutoring to get a scholarship into a top private secondary for the eldest, while still being able to afford private music lessons for each DC, as well as numerous family holidays including ones to "educationally enriching" outings to museums, galleries, theatres, etc. They may be in state schools but they are certainly not underprivileged in the slightest.
Other friends don't look like they could afford private school and aren't from a privileged background but have scrimped and saved on lots of luxuries many others take for granted to send their DCs to private school because the state school near them is failing (few pass GCSEs, cannot read or write well when leaving primary school, discipline issues, etc) or their DC simply has had unhappy experiences there.
I too have friends who have gone through the state school system and done very badly out of it, whom presumably are the ones Hunt has vowed to "help". Left with no qualifications that would help them get a job (a lone C in RE and failed English and Maths doesn't cut it) even though they speak, write and spell well (judging by their texts, emails and FB posts) and can definitely do arithmetic competently enough for most jobs when it comes to shopping together, calculating discounts and mileage, etc etc. Difficult family circumstances beyond their control as kids led to them not doing well at school. What would have helped them as kids at school was a quiet homework club (that lasts longer than the 45-60 minutes in most schools) to give them a place to do their work that home couldn't provide, a small amount of free after-school tutoring from their school - not at some posh private - not to help them beat the competition from private schools for free, but just so that they can pass a few GCSEs and leave school with enough qualifications for a job. They're hardworking, reliable people - not layabouts. The boys play football better than most private school teams I've seen. They don't need extra "mixer sessions" to socialise with "privileged kids", they just need to get enough guidance to pass.
Hunt is also wrong to suggest that private school teaching is somehow better across the board than all state schools. Teaching might not be good in some state schools that aren't run well, but in fact many state schools have just as good teaching as private ones, a few even have more well qualified or better teachers - they just have bigger class sizes.
Families who pay for private schools already pay double - both for state schools in their taxes that they aren't using and the high fees (which haven't dropped in the recession. Also, they are giving up places they are entitled to in oversubscribed or overcrowded areas to other families who might not get the school they want or even get a school place. One area near us has seen its population of under 4s triple in the last 3 years but the local authority will not build another school or increase the number of staff to create more school places in existing schools with the facilities because "we expect a lot of families will go private".
It's true that many foreign oligarchs and tycoons do indeed send their children here because many private secondary schools are indeed good ones. If Hunt has a problem with the numbers, then levy a tax on those who come in or put a cap on numbers being given visas, but don't punish local taxpayers for something the government isn't doing right.