I'm curious how you know the hospital was "happy" about it though. I ask because a close friend is a nurse/midwife who delivers a lot of babies and I find it very hard to believe that caregivers, at least, would be happy about this outcome. I also find it hard to believe that the hospital administration would reward them for this outcome; indeed, hospital administration usually inclines toward bureaucracy aimed at preventing mistakes, rather than incentives to create mistakes.

Under a single-payer system, new providers would still need to train on the job; mistakes could still happen; and the provision of care to mitigate those mistakes would still cost money. Doctors and nurses don't work for free just because of a mistake, even under a single-payer system. Administrators would still get paid too. In general, medical employees get paid to treat people; that is true regardless of how patient insurance is structured.

I don't think customers should pay extra for medical mistakes; but I also don't think providers in a for-profit healthcare system intentionally and happily create medical mistakes for the purpose of profit.