People often say that for social mobility reasons we should abolish private schools or private hospitals as it's unfair that there's "one system for the rich, one for the poor".
However I think this thinking is backward. We should encourage more gradual step-ups that enable people to slowly inject more of their own personal money into the system as they move up the income ladder and become able to do so.
For example, currently I have the choice to go to my underfunded, busy NHS GP, or a completely private GP and pay the full costs. Likewise I could send my kids to state school or go completely private.
Great if you can afford the "completely private" option.
But there isn't an intermediate option. In Australia for example, you get a govt rebate of say $30 for every GP appt. This covers a basic GP, or you could apply it towards a more expensive GP with a nicer waiting room or late operating hours etc that costs say $50 (i.e. you only pay the $20 difference, vs $0 in the free option).
In the UK you either pay £0 or £50. There's no £20 option.
In the Australian system, the health system overall gets an extra $20 that wasn't there before (from the extra top-up), and more people are able to access the nicer services (because more people can afford $20 than $50).
There's a similar argument to be made for private education. Why can you not "move" your state school funding to a private school to offset the costs? It would allow a smoother mixing along income levels instead of the harsh cutoff/separation we have today.
Eliminating inequality is completely impossible due to human nature, but keeping a harsh separation between "the rich" and "the poors" also isn't the answer. It just keeps "the rich" even more in their bubble.