The waiting time has little to do with the payment and much more to do with having enough doctors.

Where I live, the universal healthcare has a guarantee for treatment within a time dependent on the illness. If the public healthcare fails to offer the treatment within that time and if you can find treatment in the private sector, the government will still pay for it.

Both the private and public hospitals are paid a fixed price by the government by the patient, so it doesn't matter for the government (and taxes) who they have to pay.

The difference is that private hospitals can make a profit by keeping their costs lower than the fixed price, while public hospitals just need to balance. There are some overhead costs in the public hospitals, that are not necessarily present in the private sector, so it's not impossible for private hospitals to make a profit without cutting down on the service. They probably still do cut as much as they can get away with, but overall they have to deliver the same safety standards and follow the same regulations. The government audits both.

Anyway, waiting time is an issue, but not because of how it's paid.