I don't see where people like LeBron James or Bill Gates need free health care. They can literally afford any medical bill for any member of their family.
If we're going to go to government-run health care, which I think is inevitable, why not assign everyone a deductible based on their income? For example, James and Gates would have unlimited deductibles. People in poverty, maybe $100. Middle class people, maybe $5,000 or something like that. Government would pay for anything over these amounts.
Having a deductible would hopefully prevent people from running to the doctor for minor things like the common cold, but at the same time protect them from major bills. And it wouldn't be nearly as expensive as just giving everyone free health care.