I don't have any current (auto or home or renter's) insurance and need to rent a car for one week in Minnesota. The minimum, after tax, cost I can find is $280, but this includes no insurance (I even have to specifically sign that I am declining insurance). I believe that my having no insurance here is still legal in Minnesota because the car rental company has somewhere agreed to cover me up to state liability minimum limits.

So, suppose I hit somebody (and am at fault) who therefore needs $10k in medical treatment. My impression is that the car rental company (or a different company that insures them, but let me just call all of this "the company") will pay for this and then demand the $10k from me.

So, I am getting legal insurance but taking all the responsibility. It's like having an infinite-deductible policy (with deductible applying to "liability" also, instead of just "collision").

All of the above is for a rental car, but is there a way to operate a vehicle I own like this too?

Such a policy would be the cheapest, so a super-safe and super-defensive driver (who could therefore never be at fault in an accident) would want this policy. It seems theoretically possible for an owner because it already exists for a renter. Do any insurance companies offer owner policies like this? Would it be legal if they did?