As an aside, this is why buying insurance, despite being a financially bad bet (or the insurers would go out of business), actually is a sensible thing to do from a quality of life perspective.
Insurance isn't a financially bad bet. They're providing a service (not needing to maintain the liquidity of replacement costs) in exchange for a fixed monthly fee. It's cheaper for me to grow my own food but it's not a "bad bet" to not be a subsistence farmer and buy my food at the grocery store even though many people are making money off my purchase up the chain. I get to use my money and time for something more productive.
You are right for catastrophic insurance, ie insurance that covers outlays that would financially ruin you, or at least be majorly inconvenient.
Many people buy (or are forced to buy) insurance that covers minor outlays, too.
Using that kind of insurance has pretty big (relative) overheads not just in terms of money but also in terms of annoying paperwork and bureaucracy.
Some of the worst offenders are probably health insurance plans that include a fixed hundred bucks allowance towards new glasses every year. They might just as well charge you hundred bucks less in premiums and strike that allowance. (Unless in cases where that scheme is a tax dodge.)
Profit is only one part of the overhead. They also have to pay agents, adjusters, underwriters, managers, office stationary, postage, fraud investigators, lawyers, taxes, interest on bonds etc.
Similar for hospitals etc. Profit, ie cost of equity capital, is usually (but not always) a relatively small part of an organisation's overall cost structure. And the non-profit alternatives typically don't have meaningfully lower costs.