I thought that was what medic(aid/are) in the US already pretty much did.

And unless they pay enough no providers will do the procedure unless you somehow force them.

Yes, which is why single-payer systems generally do a tiny bit of research, pick a reasonable amount, and pay that.

Sure, but if you're the single possible payer, you can push that number down quite low.

Which would require providers to cut as much corners as they possibly can to stay in business.

At the end of the day their revenue is dependent on throughput of people for their facility. If they gain a poor reputation relative to other providers they will lose. Quality of care would be the only thing they could compete on.