Yeah, so... my question stands: name one you're thinking of, even if you qualify that it isn't an "endorsement". I contend there isn't one and you're fooling yourself.
* Medicare in the US (single-payer, private providers)
* Medicare Advantage in the US (government provides capitated payments to competing private companies, which then do all the administration work)
* The Singaporean system (complicated, but revolves around mandatory payments to HSAs)
etc.
Now you're saying I'm wrong because there are other socialized forms than NHS that work too. Well, OK! I submit. Use one of those.