If were going to have a public health system, people should be required to take care of themselves

On the face of it, this sounds sensible. But, thinking more deeply, who should decide the required amount of care a person ought to take? Ideas about what it means to 'take care of yourself' are varied. And consider that some citizens of this country are simply unable to take the same personal health decisions that others have the privilege to take without a second thought.

What you're talking about here isn't a public system. A healthcare system that only serves certain chosen people is not public in any meaningful sense.

A public healthcare system is imperfect on the whole, but on average, when funded and administered properly, is structured to apportion care based on need, instead of the profit motive. I think that's worthwhile, and the right thing for a society to do from a moral standpoint.