It's worth pointing out that for the vast majority of people doctors != insurance. The payer is not the provider.
The refrain of "people like their health insurance" deliberately conflates the two. People like their doctors, not their insurance. Doctors accept multiple insurance companies.
And to top it off, old people (actual old people) are already using socialized medicine. They aren't using private insurance (or, if they are, it's used to cover coinsurance/deductibles).