Yes, Medicare will be fixed by private insurers, because profit is the mother of invention. Medicare will be fixed because the insurers will find ways to become more efficient. They will cut corners when it is appropriate, which means that they will be able to provide their service at a lower price.
Medicare cannot be fixed by private companies. This is because they will be too concerned with profit. Although I like private enterprise, I believe that a system set up to help old people in their frailer years should stay out of private hands. The government has run the program since its inception, and we should leave well enough alone.
The drive to privatize government functions began with Ronald Reagan and has continued with the Republicans in a variety of venues, from prisons to schools. This problem with privatization is that government funding for a given function is reduced, since a core purpose of these businesses is to turn a profit. In health care, there are many profit-making "fingers in the pie," from hospitals, doctor practices, hospitals and other medical facilities, and the pharmaceutical companies. We don't need Medicare to be gutted by more profit-taking.
I believe the government should set and enforce price controls rather than leaning on private insurers who are likely to fail. I do not believe Medicare can be fixed this way. The rampant profit in health care needs to be stopped. There shouldn't be a concern for profit in health care, just the ability to sustain service.