Yet somehow this works e.g. in France, or for GPs and dentists in the UK. A lot of socialised universal healthcare systems socialise the insurance part but rely on private actors to deliver all or part of the service. They tend to keep the prices down by either regulating them, or having the socialised insurance provider negotiate a fee structure on the basis that the insurance provider can offer the lowest amount that gives sufficient capacity of the required standard.

E.g. in the UK, general practitioners negotiate as a block, and individual practices can take or leave whether or not they want to provide services for the NHS or private services. The NHS needs to offer contracts attractive enough for enough practices to take NHS patients and ensure continued recruitment, but no better. It's a market, but skewed by a very large actor with a strong interest in keeping prices down.

If they "just" let people shop around without regulating or negotiating prices (and leveraging the volume of business), then it would be a problem. But having the consumer participate in driving costs down is demonstrably not necessary. On the contrary I'd argue it's counter-productive, because when dealing with people with less money it creates a risk that people gamble on options that cost less in the short term at the expense of creating higher costs in the long run.