Not necessarily saying our system is wrong, just that the systems being so different can make people confused. :)

The alternatives involve pushing some of the financial costs of driving onto people who create less of these costs

I mean, our current model does that. All insurance does. You pool costs with the expectation that most people won't need as much as they put in.
A significant amount of our costs are based on statistical, not individual, risks. A 23 year old male is going to be charged more for car insurance regardless of their driving record.

A better system might just be universal car insurance via vehicle registration. Bigger pool, easier to accommodate people who can't afford adequate coverage, and it better ensures everyone's cost is covered.
It's also nice to not have a law forcing people to buy a product from a private company.