From living in different countries, my feeling is that a public-private sector is probably the best. I was ok with Germany's public-private healthcare, while it has its own problems, I think it worked quite well. I loved Switzerland's (private) but I do not think it is easy to translate the system of a small rich country to a large less rich one. I did not like the Italy's one, it is better than UK's for small things (e.g., visiting your family doctor) but I had mixed experience for more serious stuff (great first class treatment for my brother who had a life-threatening infection, but abhorrent for my father who had cancer). I hated US one, while it was great personally for me (top-tier health insurance), if you have no insurance you're literally left to die or need to go into massive debts.
I have no idea what is the best, but I am not sure if a fully public sector works, at least not without making massive concessions in other parts (lowering the budget somewhere else, or drastically increasing the income tax*). I think NHS needs a fundamental change, not patching it. I do not think fully private is a solution (at least not for many), and even if it is, it needs to be extremely regulated, unlike in the US. I am also not sure if fully public is a good solution though.
* I do not think that is also a solution cause it is already quite high. Taxing more the super rich while appealing in nature IMO does not work simply cause then they leave for Monaco. Increasing the corporate tax looks the best bet to me, I do not see much arguments against it.