The answer to every question like this in the US is "because the wrong people would benefit".
In the past (and still today but less so), that literally meant black people. Now it mostly means poor people of all races.
It's been shown over and over that it's practically an American cultural value to accept lower benefits to yourself in trade for denying benefits to what you see as an undeserving group.
I’m pretty sure EVERYONE is covered under the public system, a levy is added in the car rego fee that is the funding mechanism. You don’t need to be a driver to be covered.
In NZ we are not required to have third party medical cover because everyone is already covered by yearly car registration fee.