So your complaint is that the young subsidize the old in a public system, and your solution is a private system that somehow doesn't raise rates on high risk (older people) and to use the young...to pay for the old?
You also ignore that you can't switch and magically have 30-40 years of float for old people currently receive healthcare, so you have to keep the same system in place until they are gone because insurance companies would instantly go bankrupt under your plan (since they have yet to build a float but have payouts instantly), so now young people subsidize old and have to pay for their non-subsidized future so they will basically have to pay double. Or do you plan just leave old people out of the public system? Pretty nice demographic to just ignore in your plan.
You also ignore that you can't switch and magically have 30-40 years of float for old people currently receive healthcare, so you have to keep the same system in place until they are gone because insurance companies would instantly go bankrupt under your plan (since they have yet to build a float but have payouts instantly), so now young people subsidize old and have to pay for their non-subsidized future so they will basically have to pay double. Or do you plan just leave old people out of the public system? Pretty nice demographic to just ignore in your plan.