What clause in the US Constitution makes this a federal issue? Regulating autos is a state issue.
Interstate Commerce clause. In tandem with State regulation. The States are responsible for registration and licensing vehicles. The Feds are responsible for vehicle issues that affect multiple states in parallel.
As for "conflict of interest" by Musk, as long as the agency cuts are made across the board and not directed solely at the division that investigates car crashes, collects statistics, and such, I don't see any conflict. (But am willing to be disabused of incorrect notions.)
Interstate Commerce clause. In tandem with State regulation. The States are >>> Which clause directs the regulation of personal automobiles used to exercise the right to travel. Not doing any commerce in the car.
Musk is not making these cuts. The president is in collaboration with the agency director.