My guess is that court shoes are not as spongy and padded on the bottom as running shoes, so it might induce a lot more vibrations and such over long distances. Obviously, when we're playing badminton we need to change directions really quick so it's not feasible to have a squishy sole that has little lateral grip. Also, I think I would be pretty sore all over after running 10k in running shoes too
How about "no" as the answer to your question... because we're not that silly to do that in the first place?
No, each type of shoe is designed for a specific purpose or they would not be made.... Running shoes really narrow at the front and also curve upwards to help with your stride. If you look at the side profile of running shoes you can see this. Badminton, and other court shoes, are made to be flat so that you can get as much surface contact as possible for those quick changes in direction. Running shoes are specifically designed to go in one direction (forward) very well.