Most Americans experience first hand how horrible health care here is. No one can get upset at whatever little details are involved, however 'cause it's beyond them. Especially, you won't get people excited about more competition in a field where markets and competition seem neither logical nor have shown any benefit.
The complex referral system you describe basically results in turning regulated monopoly behavior into unregulated monopoly behavior. And this is the upshot of breaking up the original, regulated monopoly Blue Cross/Blue Shield, which provided adequate insurance in America's "Golden Era". Which to say that, imo, "restoring the free market" isn't a useful action in this mess. State controlled medicine is essentially the only solution to the already partly collapsed US health care system.