No, it is certainly NOT true!
That's why there can be bazillions of old unregistered cars and trucks in garages, barns and back yards, and you sign a thick stack of wavers absolving everyone in sight before anyone lets you on a race track.
In any case, there is no Constitutional right to drive a car. Even if Congress went to the 'rats and a 'rat President signed this, it's unconstitutional. First, she needs to get rid of that pesky Bill of Rights. (While we're at it, how about repealing the 19th Amendment?)