It definetaly helps for overall cardiovascular fitness.
I see good running (which I'm far away from) as kind of foundation skill. You don't get fast feet on court by just running fast 5 miles. But you build a solid foundation (in terms of work capacity) on which you can train to become faster on court.
So there is def a correlation between good footwork and endurance on court and your mile-time, but certainly not a direct one-to-one relationship.
The same holds true for e.g. weight-lifting skills. Clean&Jerking 200lbs will not make you a fast and explosive badminton player, but chances are good that an athlete who can C&J well, will have some kind of transfer to badminton.
The goal is to become a well rounded athlete as some kind of base which you can build on. This will help prevent injury and over-exhaustion as well.
But to be honest: Badminton specific skills have to be acquired on top.