Well, yes and no. It should probably be a state-run system or at least a heavily regulated system where the companies are limited in their profit making. No idea how an ideal system could look like. Here in Germany there is a two-fold system, which a generic public health insurance (with several companies offering those insurance services), where every employee pays a certain percentage of his salaries as insurance fee (actually the total fee is split 50/50 between employer and employee). Service is rather basic, but sufficient.
And then there is the possibility to get a private health insurance contract, if your income is above a certain level, which interestingly is (for the most time) lower than that in the general public insurance, but service is much better (e.g. you usually get doctor appointments much faster if you are a “private patient”). The only downside is that you don’t know how much you will have to pay when you get old, and once you are out of the public insurance you can not go back (only if you income falls below the private insurance entry level, which is rather unlikely).
It’s not ideal but it works for the most part and with some exceptions (like new teeth, where you have to pay a substantial part by yourself) you don’t have to be afraid of any health problems, operations or whatever, because that’s all covered. Those insurance companies are treated like public service companies and prices for medication and medical (doctor) services are subject to agreement between the government and the medical associations representing doctors, hospitals etc., but I guess those companies still make profits and the doctors have good earnings.
I get your point, but even with a certain level of protection you’re probably still better off than with no protection at all. However, the system should also not be based on profits and shareholder value, that’s true.