people choose what services they use supplied by private companies all of which will be competing for business
You'd think that privatisation would improve efficiency through market competition, but does that ever happen in practice? In my experience, private healthcare companies charge the maximum they think the market will bear. Also, don't forget that an insurance-based (or hybrid) system, requires extra layers of administration and extra middle-men in the form of insurance companies.
Do you think private fertility clinics in the UK offer good value for money? After all there are plenty of them, which should keep prices affordable, right?
What about private dental practices? At my local practice, a scale and polish by a dental nurse costs £85 and they still don't have any appointments available until June!
Do you think that vet practices in the UK offer reasonable rates? These days, many owners have pet insurance (and some insurers pay the vet directly), so the combination of market competition and the expert negotiation offered by insurance companies ought to keep prices low. But does it?
US private healthcare companies have been desperate for years to run the NHS. They wouldn't be so keen if there weren't enormous profits to be made.