"and healthcare, for all it's importance, is nowhere near being as important as, say, free political speech."

I assume from this statement that you are either a) healthy and/or b) fully capable of paying for all you and your families medical needs - both now and in the future.

"I'm unaware of any right that involves making other people take care of me by producing medicines, providing services, food, housing me, and tending to my disabilities."

Australian doctors are highly paid. Their salaries are paid by every Australian tax payer whether healthy or sick. By sharing the burden all of society benefits.

You would be much more inclined to agree that healthcare is a basic human right if you were both sick and unable to afford treatment.

The standard of living in Australia is as good or better than in the US. My medical costs are $0. Regardless of whether I am visiting the doctor for a flu shot or having life saving heart surgery.

Better systems are possible.