Insurance was called Major Medical...and only used for emergency needs.
(Agreeing with you) Surprisingly, that is exactly how car insurance works today: it will give you money if you crash the car. That's it. Routine maintenance? That's your expense. Car insurance does not pay for oil changes, timing belt replacements, petrol fill-ups, new tires, etc.
Health insurance should also not pay for the equivalent.
The counter-argument: "A car is a thing, health is everything. If you make routine health cost money, then people will stop going to the routine visits." I suppose there is some truth to that argument; there will be some edges-of-the-Darwin-pool who value fiat currency over their health, and they will reproduce appropriately. Similarly, one could argue that if the insurance company does not bathe you and wipe your bottom, then you might stink, and smell. But nobody is making that argument, and in that light, doesn't the counter-argument seem a little bit more ridiculous?
Of course the other part of the argument is exactly as you stated: prices will go down when there's competition, so the foresighted people understand that the routine visits will cost much less. And eliminating middle-men from any transaction usually benefits everyone else involved in the transaction (except for some levels of volume, scale, and shipping, generally none of which apply to doctor's visits).
And the best part of your argument? People might spend some of their annual medical budget on reading up on things like Jin Shin Jyutsu and EFT which can be done in the privacy of one's home, by one's self, both of which help to heal the organism and prevent the need for modern medical treatments.