Insurance: pooled risk.
Loan: borrowing against future income.
With insurance, you may face higher premiums if you make a claim. But you won't pay anything against the claim itself. With a loan, you are responsible for paying the claim. You pay some of it with past income (savings) and some with future income (borrowing). But in the end you pay the whole thing.
Insurance can pay more than your entire future income. Instead of you paying alone, you and everyone else who bought the insurance pays. Once the event happens, you can stop paying if you don't mind not being covered.
A loan has to be less than your expected future income. Otherwise your creditor wouldn't give you the money. Once the event happens, you start paying.
The question basically comes down to how you want to handle risk. Would you rather pay when you may not need it (insurance)? Or would you rather wait to pay until you know you need it (loan)? On average, you'll pay about the same either way. Each person pays less for insurance, but they pay it more often.
Consider the following circumstance. You're twenty-five and expect to live to age seventy-five. When you are thirty-five, you or one of a thousand other people might have to pay \$100,000. An insurance company offers to let all thousand of you buy insurance for \$11 a year. So on one side, you pay \$110 over ten years. On the other side, you pay \$100,000 over forty years. Would you rather a guaranteed \$110 or a possible \$100,000? Assuming an interest/inflation rate of zero.
Of course, real insurance decisions aren't that well bounded. But the principle is the same. You pay a small amount ahead of time to be assured that you won't spend a large amount later.
I was thinking that it would make more sense to save the premium money and use it to pay part of a loan to cover the losses if it happens. That way one would save the trouble of paying insurance and never using it and make better use of that money.
That would work great if the event was guaranteed to happen to you. So you have to pay \$100 in ten years, guaranteed. Or you could pay insurance for ten years. Say \$11 a year. Then obviously, you'd be better off saving \$10 a year than buying insurance. At that level of certainty, there is no risk to offset by pooling.