Many free market thinkers immediately put up their hands to private. All well and good if you have the means and the insurances have no very small printed clauses. But universal has its own problems. Abuse and misuse and free-riders if there are no clauses relating to pay-in-get-out. Also it leaves those with less wealth less well off and those with less wealth or none in the same boat. So what's to be done?
Firstly one has to decide which of three things one is in favour of:
- Universal paid by tax.
- Private paid into insurance and or saving schemes.
- Means tested.
- [a] Minimum universal payment
2. [a] Complete private funded and provision of services
[b] An opt-in clause to 
3. [a] Every benefit is means tested to account for wages and savings
[b] Every benefit is means tested to account just for wages
[c] Every benefit is means tested to account just for savings
[d] An opt-out clause to go into private funded and provision of service
At the moment in the UK there is only one real choice and that is universal payment, with private insurance an option as well if one can afford to do so.
My choice would be since I live in a welfare provided for State 3[c] & 3[d].
That way people have some choice. Private providers would need to provide according to the market and not remuneration from the state. Its an idea at least.