Cailin - I don't know. I don't think it's a step that could be enforced overnight. I think it would have to be worked up to in a piecemeal fashion. Rome wasn't built in a day and all that.
To start with maybe a huge state-sponsored provision of bursaries and scholarships for children from ordinary/disadvantaged backgrounds to enable them to attend top private schools, with incentives (e.g. tax breaks) for such schools to take such children.
A mass restriction of university places back to the brightest and best, with places fully paid for by the state but available only for those who achieve good qualifications no matter how wealthy their parents.
While the old grammar schools weren't without their problems, they were good levellers in some ways and did a lot to encourage social mobility. Learning the lessons from what worked in those establishments would be a good place to start.
An end to internships and apprenticeships that no self-funding individual could ever hope to do because of the minimal/absent wages. That goes as much for the trades as it does for internships in trading companies.
MrsMorton - the problem with that little analogy is that the underlying premise is that everyone with money has worked hard to get it and that those those who work hard will always get money. There are plenty of people out there who work their guts out for very little. Your story also takes into account the very real problems responsible for a person ending up homeless in the first place - problems that may be social in origin and mean that person is incapable of leading a normal, self-sufficient life. Are only good happy workers worthy of basic human rights? (the proportion of homeless people with MH issues is off the scale)
Despite that, a recent study showed that those with lower incomes gave more to charity as a percentage of their income than those with greater incomes.