Insurance is a good model for things like car insurance and home insurance because the duration of a claim is very short - one never has to change one's insurance provider for reasons beyond one's control during a car crash or an earthquake. But a medical condition can last for many years or even a lifetime, during which there can be many individual claims.
Once you have a medical condition that is likely to take many years to resolve and be extremely expensive, you're beholden to your insurance provider, since you are a liability for them. Should you be allowed to change insurance providers? Okay, most of us can't change providers anyway without spending a lot more (to avoid paying tax on our health premiums we're stuck with our employer-provided health plans) but suppose that problem was fixed?
Let's suppose the answer is no, you shouldn't be allowed to change (at least not without taking a cripplingly large increase in premiums turning it from insurance into a payment plan) - no business should be required to take on a customer that is guaranteed to lose them money. Then the "free market" aspect of health insurance is denied to those people who need it most - it's not a free market when you can't make a choice.
So let's suppose the answer is that you should - that insurers should not able to deny access to insurance based on pre-existing conditions (which really ought to include genetic predispositions). Then insurers have to factor in the costs of such patients to all their customers' premiums. That doesn't sound so bad at first glance but consider the consequences - suppose that insurance company A had low premiums but covered very little and insurance company B had high premiums and covered everything. Then all the healthy people get insurance from A and switch to B when they get sick. B's prices then rise to cover the cost of their coverage and what you essentially end up with is equivalent to not having insurance at all. This means that government has to regulate what coverage is provided as well as who it is provided to. All insurance companies will then provide that level of coverage but none will provide any more. We've gone from having no choice to having no real choice. Either way we fail to have a free market for healthcare.
Socialized medicine aka single-payer healthcare aka politically controlled medicine has problems too. In the UK, different districts decide which treatments they will offer, resulting in a "postcode lottery" for some treatments. Waiting lists of years are not unheard of for treatments that are not imminently life threatening, and doctors and nurses are overworked and underpaid compared to their counterparts in the US. However, the average Brit pays about half what the average US resident pays in healthcare costs and gets better coverage on average. There is also no worry about medically-induced bankruptcy, and no worries about one's insurance company refusing coverage (due to a "pre-existing condition") once a serious medical condition is found.
I disagree that single-payer healthcare gives patients an incentive to over-consume. It should not be the insurance company that gets to decide that - doctors should be ones to decide which treatments are medically necessary. And if some doctors are over-prescribing compared to other doctors in their geographical area or speciality, that is something that should be investigated.
Another criticism of single-payer healthcare is that providers have no incentive to compete on price - the government will have to pay whatever they ask for critical treatments. But there's nothing special about single-payer healthcare there - there's nothing stopping the government from acting the same way as insurance companies to do to keep a check on prices. In fact the government has more power than an insurance company there - the insurance company only has the power to drop an overcharging provider from their preferred providers list while the government has the ability to take away their medical license.
Yet another criticism is that if the government gets to decide what treatments are on offer, people don't have any choice. Well, most of us don't have any choice anyway under the insurance model but at least with the government you can vote them out if they aren't doing a good job with healthcare. Yes, socialized medicine means a government bureaucrat between the patient and the doctor, but better an elected bureaucrat than an insurance-company bureaucrat that one has no control over.