> If we hit a point where nearly all people were on Medicare, we'd be bankrupt anyway.
No we wouldn't, we'd be in a much better position and we'd be paying much less for healthcare. Medicare has far less overhead than private insurance and it'd be the logical move to establish a far more effective and cheaper system of socialized medicine.
But continuing with your theoretical environment: no, they would consider adding new patients until they had a critical mass and then stop accepting new patients.