I thought the point of insurance companies was that the customers wouldn't pay the full cost.

The point of insurance is that the customers pay the full cost. (And a little extra). It's just that customers share the full cost of all the customers, not their individual costs.

The point of insurance companies is to hedge against financial scenarios that you cannot otherwise plan for.

There's no world in which adding a middleman for a regular and ongoing service lowers the overall cost of that service.

It's passed on in aggregate, spreading the burden. Except now the burden gets heavier for all because some companies find a way to leverage the most vulnerable.

So, there is such a thing as a free lunch?

The funny thing is you're being glib, but Markowitz literally is quoted as saying diversification is the closest thing there is in finance to a free lunch.

Diversification? An insurance policy is a wager made with one party on a series of closely related events. There's nothing diverse about it, from the consumer standpoint. Some diversification is possible for the insurance firm itself, although there are limits to that so most firms purchase reinsurance.

Of course it's diversification. Instead of being short your own illness only, you enter a swap where you receive coverage for your own illness (thus being flat on that clump risk), and pay a small floating leg covering (a small proportion of) everyone's illness. You have much smaller variance, but the same expected return (minus administration costs) - just like any diversification.