What solution that provides universal coverage would you advocate?
1) Leave the private health insurance market completely alone.
2) Lower the age of eligibility of Medicare to zero. Do that gradually (five years per year, for instance) if politics require such a compromise. Raise Medicare taxes as required to cover the increased number of people.
Change Medicaid eligibility so that anyone under 18 is eligible, until such time as they're eligible for Medicare (see caveat previous para.).
Done. Net effect should dramatically lower the cost of private health insurance, since Medicare would cover most (if not all) common problems, leaving private health insurance for edge cases.
Adjust Medicare taxes as required to pay the bills.
Note that this isn't quite a National Healthcare System (see UK), but it could easily transition into one later if it works reasonably well.
Done. Simple law, expanding existing program, so unlikely to meet as much opposition from fanatics. No odd cases like "he makes $XX, so he gets 80% subsidy on his health insurance, she makes $XX-1, so she has to pay 100% cost of her health insurance.