The HW Mustang is not a drift car but a thoroughbread race car and like all thoroughbread race cars in this game it lacks the drift suspension upgrade. It sucks, I know, but it is what it is. Most annoying is that even the Caterham doesn’t have drift suspension.
It’s racing slicks, not drag tires. Some of the thoroughbread race cars have racing slicks, opposed to the normal cars which have threaded, street legal, “race” tires. Try putting drag tires on a car (in the game) and see what happens, it becomes completely undrivable in corners. Also do a google search and see how many drag cars that have massive front splitters.