The alternative is that cars are prohibited because we've decided not to insure the risk and their owners definitely can't be relied upon to just happen to be able to cover the costs when, inevitably, they are incurred.
I'm OK with "all private motor vehicles are prohibited" but you need to be clear if that's what you want
To extend your analogy slightly (while recognizing the actual meaning), if you go to an axe throwing venue (range) you dont have to carry insurance, but the range does carry insurance.
If you go into the wilderness where axe throwing is allowed, and you maim or kill someone, you are personally liable to be sued or prosecuted.
If you throw an axe where it is not allowed, you are also personally liable criminally and civilly.
I'm OK with "all private motor vehicles are prohibited" but you need to be clear if that's what you want