I run sprints in training and distance for fun. Most of my former coaches ran 3-4 times a week, my last coaches wife is a competitive runner, you will never see any of them using any running shoe on the court for serious play or practice. Personally I run in the CC Rides and just got some Charge RCs (AWESOME shoe) but also got a pair of Minimus Trails from my gf, I'd not be caught dead on a tennis court in any of those unless I was running sprints. I do have experience with a lot of different shoes and I can tell you that there's a reason tennis shoes are TENNIS shoes, they are built to protect your feet in the sudden starts and stops and changes in directions.
I have rolled my ankle, running sprints in the Rides and in the Barricade 5, the roll in the Rides were worse. The way I play I put my body under a ton of stress, I'm off the ground on just about every forehand and will try to run everything down, because of this I need shoes that will keep me safe and stable. If you just bat the ball around feel free to use slippers for all I care, but if your an actual competitor you should be in specific tennis shoes.
And if you think you can't roll your ankle barefoot, try street rugby or football. Not hard.
"IF YA AIN'T REPPING 180 ON THE BAR BENCHING, YA WEAK!!! ARGHHHH ARGH ARGH!!!"
-Scrub frat boy