Your example is an edge case. Presumably sick people were once healthy. When people are spending their own money, they become a lot more discriminating.
Also, if you're looking for treatment after you're sick, that's not health insurance.
I'd tend to think the majority of healthcare is sought after something happens - car wrecks, cancer diagnosis, sports or exercise injury, stuff like ear infections or strep throat...
Healthy people without concerns aren't looking for treatment, they're visiting their PCP once a year.
Also, if you're looking for treatment after you're sick, that's not health insurance.