>It would be different if it were the same exact group of people and the same budget who are in charge of plane safety, as those in charge of the car safety
This is exactly the case. In the US we have the Department of Transportation that governs automobile transportation as well as air travel (and also trains and boats).
But the FAA and the NHTSA are different agencies under that. People who work at FAA can't switch tasks and stop working on plane safety and start working on road safety. With time of course either agency can be funded or defunded but that can be done independently. Just because NHTSA gets more money or resources doesn't automatically mean FAA will get defunded and vice-versa.
This is exactly the case. In the US we have the Department of Transportation that governs automobile transportation as well as air travel (and also trains and boats).