Governments at both the federal and state levels have so many regulations and requirements relating to private health insurance companies that seniors would be hard pressed to find a comparable plan on the private market, especially if they have some health issues. The one thing about Medicare is that it basically has no cap on enrollee spending. If you want to make private health care affordable to seniors, you have to remove all of the regulation or red tape, or it will be simply unaffordable. Seniors don't need private plans that cover things like abortion or STD screening, for example.
The elimination/phase-out of Medicare has to come with significant repeals of federal and state legislation governing the health care industry. This would allow for more competition in the market and allow for more companies to offer catastrophic plans. The system we have now is broken -- even the private system, because it decouples costs from benefits. If you are covered by your employer under a typical private health care plan then you have almost no incentive to reduce your health care costs.
A few years ago I read an article by a guy who said his dad had come up with a fantastic idea for National Health Insurance. It was very simple. It went something like this: The government would underwrite catastrophic health policies with an $80,000 deductible and then the individual citizen would pay for a health insurance policy up to $80,000. The theory was that all of the insurance companies would be happy to underwrite inexpensive health policies that maxed out at $80,000. I always thought it was an interesting idea. Point being that there are probably hundreds of great ideas about how to manage health care costs if we could just get our legislators off the dime to take a look.