Feels like a bad idea to me. Why not just tax the hell out of the private treatment to make sure the public treatments stay funded?
I guess the concern is similar to the public vs private school situation, where some areas have bad public schools because everyone with any money chooses a private school and has no incentive to fund the public system.
Still I find the idea of government-only medicine to be scary. In the past we've had politicians who have decided that things like abortion, stem-cell therapy, and medical marijuana were all moral evils. I am glad people had private options to get those things.