A single buyer negotiating with multiple sellers can force prices down to cost. Whether it actually does so is a public choice problem.

Well since single payer typically refers to the government being the single payer, the bill goes to the government because you the insured are not the payer of the bill.

The bill goes to the government, who gets its funding from the people. This in no way stops any systemic over-billing that may be occuring.

Well, in my country, the government would laugh, and then only pay a standard amount.

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.

See: Japan where the government decides the price of procedures.

On national health insurance (monthly cost depends on your salary but for an average person it is a few hundred bucks per month) the hospital pays 70% and patient pays 30%.

It means basic visits to the doctor or dentist are very cheap here. Like $20 for consultation + medicine. ER+X-rays and MRI (appendicitis, sigh) was a little bit over $100.