Do you think of the fire department as “some sort of public insurance”?

Yes, it can be viewed as such! Things can be viewed from many angles. Another one that're not called insurance but kinda are, are flat prices for things whose price actually varies. Like electricity prices: they're often based on a spot price in an power exchange, so if you as a consumer choose to get a flat price, you're essentially paying a premium.

Insurance is a pretty great industry and good fit for markets if only there's healthy competition. If there's no fertile ground for competition, it's one signal that perhaps the public sector should handle that one.