I will say that I recently had to deal with the medical system for a family member, and he was on decent private insurance. We actually would have received better care at a lower cost if he was on Medicare instead. N=1, of course.

I'm pretty sure that Medicare has been shown to be a more effective negotiator than the private insurers and generally operates more efficiently, but I may be misremembering.

Disclaimer that I don't support medicare for all myself, I'd much prefer a model closer to the NHS, with caveats (federal funding, county administration, state oversight).