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.
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.