Oh, so you mean, "only if you don't live and work in one of a prohibitively small number of major metro areas in the US".
For all intents and purposes, auto insurance is effectively mandatory for most Americans. It's for a similar reason that many states revoke driving privileges for parents who don't comply with child support.
You have to own a car, though. The state makes you buy car insurance because you can do a lot of damage to other people and their property with a car. They own the roads, so they're responsible for making sure you can pay for that damage.
Texas, for instance, doesn't actually make you buy insurance. You only have to prove that you can be financially responsible for the damage you could cause [1].
Why is this relevant? You made the claim that "The sole purpose of the mandate is to make free-riding on the insurance market unlawful"
The medically uninsured only pose a financial threat to themselves. It's only through laws that make taxpayers responsible for the medical costs of the uninsured that they pose a threat to the rest of us. This mandate won't change anything in that respect. It will just make the line from taxpayer money to person who can't pay for his medical bills harder to follow.
The laws making the rest of us responsible for the uninsured aren't really optional, so it's probably better to just reason about this problem as if the uninsured were just intrinsically a financial risk to us.
For all intents and purposes, auto insurance is effectively mandatory for most Americans. It's for a similar reason that many states revoke driving privileges for parents who don't comply with child support.