In Australia I just take my blood test form to any pathology place and they do it for free (for me) and bill the government a set price from the medicare benefits schedule.
I lived in both systems. In a single payer system, the state essentially decide what is allowed and what is not. And with the state as a single payer, they also go back and forth on price with the hospitals.
It still is a better system overall but there is no places where you can just spend as much as you want on healthcare without some type of centralized supervision.
That and there is often a 'gap' that needs to be covered for GP and specialist services, although that tends to be balanced out by much cheaper prescription costs. (Prescriptions in Canada for example easily cost 2X as much).
However, Australia has a two-tier system where you can buy private insurance cover that can cover the costs of gaps and allow you access to private hospitals. This insurance is much cheaper than the equivalent US versions.
Usually if the state decides you are not eligible for something (for example, some examination), you can just pay and get it anyway.
I had a test recommended once that was not covered, but my Dr explained this in advance and the cash price to me was $110. There are no 'surprise' denials after the fact.
this is the crucial difference.
It's absolutely fine (and required even) that single payer public healthcare doesn't cover every conceivable thing under the sun. It should cover the most common, easily scaled and mass produced items.
For the remainder, the patient should be told and know what to expect price wise - private or self-paid etc. And this also allows competition between entities offering this thing that's not covered.
Having opaque and unknown pricing (until after you've done it) is basically a form of highway robbery.
Of course there is dentistry, which is a complete nightmare... people trying to do their own extractions with a pair of pliers is the sort of thing you used to associated with the US but I've actually met some people who have tried that due to how poor NHS dental services are and how expensive private treatments are.
You're right about dentistry.
With any insurance you get 2 cleanings every year fully included and most routine fillings are almost completely covered. Easy to find appointments everywhere
My experience in Europe has been that it is super difficult to schedule anything. Waiting time of multiple months for new patients.