I can't figure out how to do health care better than a government.
Thats right! and you are only human.
Governments also, are only made of humans and have the same problem.
Being a government doesn't grant special 'figure it out' powers, only the ability to
take more resources.
Governments can't do health care better than free markets.
Lucky for governments they can make competition illegal or extremely expensive.
The Government then does not need to worry about 'how to do healthcare better'.
Nowhere in the universe can you do anything without cost. It cost me 3 calories to eat this sandwich.
Government anything, has a greater cost than the free market solution, simply because they have no incentive to 'do better'.
I assume the fix would be blockchain based insurance providers and hospitals. But what happens to the poor? Is the libertarian thing to do just let them die off?
blockchains can potentially save money anywhere they are employed. In the case of insurance, nearly the entire industry could be replaced. Hospitals still need doctors and facilities.
Libertarians also, are just humans, they have no greater or lesser feeling for the poor and missfortunate. In a libertarian state, everyone, not just libertarians, would keep more of their earned money and waste less on government. So everyone, not just libertarians, could spend that money in care of the poor, or whatever their hearts desire, at the greatest efficiency.
Does the free market work with health care? Isn't there an incentive to keep you sick or even make you sick?
No matter the type of government.
People will always have these incentives for their own short sited gains.
The best solution can only be found by removing all barriers, freeing the market to offer all number of options.
Then it is
your choice to listen to the doctor who only wants to make you sick.