There's no need for any of those services to be directly publicly funded to work.Oh, I think we are actually very close to agreement here!
Allow the experts to deliver public services within political interference - you should become a civil servant! Because we are expert at delivering public service; at ensuring the service is there for the needs of the people and taking a holistic view.
Irrespective of views on capitalism and market forces, private company cannot prioritise a holistic view ahead of profit - they exist for profit. Nothing wrong with that, except where it's an important national infrastructure service being delivered and operating in a competitive market.
There should be no important services to which cost is the barrier - as a society we should create a decent baseline and also allow private industry to offer better services for those that choose to pay. Similar to how we do with dentistry - although that should be free at point of use for NHS patients imho. Personally I choose to pay three times the price for dentistry, and really like the results. But someone that would prefer to spend that money elsewhere and only receive check ups and essential works should be able to access it.
Germany doesn't fund its healthcare system the way we do and it's far superior. My experience in the US is that their hospitals are significantly better too (although aligning their system towards Germany's would make it more universal).
Private dentistry is the correct option. But you shouldn't have to pay the tax for the public dentistry you're not using if you opt out. Same for schooling and healthcare.