Most tire shops have a profit motive rather than a "provide tires" motive. It's true that tire customers aren't "captive", but I don't understand which sense of that word you mean to apply in a relevant way to health care consumers.
You don't understand how health care consumers are captive? Health care is literally something human beings cannot live without. The quality and cost of the care is far less important than its availability, because it's literally a matter of being healthy and surviving, or becoming or continuing to be ill, deteriorating and then dying. Healthcare consumers, in other words, don't have real choices in where and how they obtain care. This isn't the case with tires (or almost any other non-essential good or service).
