Labour passed a motion at their conference in Brighton this weekend which outlined a commitment to "integrate all private schools into the state sector" in the next manifesto.
The motion adds universities would have to admit the same proportion of private school students as in the wider population, now 7 percent.
The delegates made the remarks as part of a dramatic day which saw Ms Rayner, shadow education sectary, pledge that Labour would scrap the "tax loopholes" that private schools benefit from in its first Budget.
Assets are often used by private schools to generate income, either to help maintain their independence, or, along with the tax-exemption charitable status - which Labour also vows to abolish - allows them to provide bursaries and scholarships to working class pupils.
"Whether it's Shami Chakrabarti sending her son to a private school, Emily Thornberry sending hers to selective schools, or Jeremy Corbyn putting his own political views ahead of his wife and children, the Labour Party are in no position to lecture anyone on what is best for their family".
"This huge leap forward is a testament to the hard work of grassroots Labour members and the ambition and determination of Angela Rayner and John McDonnell".
He told BBC Radio 5Live: "What I think we'll find is that for the large number of people who would be, at the moment, spending money on private education, that money actually will go into the wider economy and stimulate the economy".
After Labour had banned establishing new academic-attainment selective grammar schools in a previous government, there is the question of where Labour's policies of stopping school choice or making illegal so-called "unfairness" in the education system would end, given British parents also pay privately for tutors, music teachers, and other non-State controlled extracurricular activities. "They are the disease that infects our educational system".
He said: "If implemented, it will be an act of unprecedented vandalism".
He added it would "threaten" the jobs of teachers and support staff, noting: "Empty promises of transferring over to a comprehensive state system ignore employment law and basic humanity".