In my state, I pay $15k/year in school taxes, yet I have no children. I pay $1000/year in property taxes to support my city's library, yet I don't have a library card. People are taxed for lots of things they don't actually benefit from. I don't think we would need to force rich people to use the plans. If they want to buy medical services from private doctors, sure we can let them.
The issue then becomes more about allocation of resources (how many doctors are available to be seen on the public system vs. only available to self-pay customers) rather than the issue being about how to collect taxes.