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It's a n-tuple-whammy. Co-pays and deductibles discourage early diagnoses by preventing most people from visiting doctors for issues that aren't serious. The painful process of claiming the small amounts that would cover preventive care discourage people from getting it even if they've seen a doctor and been told to. Preventable conditions balloon into things that are very expensive and unpleasant. This process takes years though, and there are many, many insurance companies, plus people change jobs and coverage frequently. Odds are, the company denying preventive care won't be on the hook for the heavy costs down the road. If they are, they have ways to wriggle out of it. Also, with more people waiting until conditions get bad before seeking treatment, the whole system consumes more resources to keep people less healthy, on average, than if the system pushed preventive care effectively.
When governments assume responsibility for medical coverage, incentives start to be much less perverse. You pay taxes while you work. You work while you're healthy. So, the government wants you healthy. The government wants to do that as cheaply as possible, so they're incentivized, not just to provide preventive care, but to make sure you take advantage of it. I live in Canada. I have a genetic condition that could cause me problems if I don't get preventive care. One of my relations had those problems, so they tested her and found the problem gene. The government then tasked a genetic counsellor to seek out everyone related to her. I had the gene, so they now book me for the care I need to manage it. I don't have to pay for this. I don't have to fight insurance companies to prove I need preventive care. I just do what they ask me to do and I get to live longer. This is the kind of stuff that happens when the incentives are right.
Insurance companies need to be cut out of the loop entirely. They know they're what's wrong, so they employ expensive lobbyists. Americans need to realize this and start making it an election issue.
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EDIT: TIL that preventative and preventive are used interchangeably in the context of medicine, so I changed my very inconsistent use of them to just one.