Healthcare wait times vary a lot country to country: https://worldpopulationreview.com/country-rankings/health-ca...

27% of cases take one month or longer to get to a specialist in the US. Compare that to 61% Norway and Canada and 0% in Spain and Chile, all of which are single payer systems. That figure is at 25% in Germany and The Netherlands, which are both mandatory multi-payer systems (a mix of private and public insurance with much stricter limits on out of pocket costs compared to the US). It's not obvious that single payer systems lead to universally better or worse wait times.