It should never be shopping in the first place. I live in Canada, my family has needed the hospital and the ER a few times and we never ever had to even reach for our wallet.
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.
