Remember, you have no autofocus, so recording games like football, basketball, etc will be manual focus. Rolling shuttter will be far from your worst problem.
You are right, i was wrong about the unpredictable games (almost every team game), but for example in formula one, u know exactly where the car will go, and you have bright sunlight, u easily can go up to around f11 (over that, diffraction will effect your shot much more cause you are in zoom mode), with what you get a pretty usable DOF even if you are at 500mm. Or set your focus at a fix position, and wait for the car, at 5x zoom, it will still look awesome
Or gymnastics (bars or pommel horse) where the subject moves only in a small area.
Or a better example: If you want a fast action shot in your short movie, u take the 10-22, set it to around 15mm, go for a 3x-5x zoom, and BAM you get a much better, and a much more correctable rolling shutter (i don't say that it will disappear, cause it wont, but you will be able to fix it easily.
Another interesting question, that can you do a 10x zoom with a apsc sensor without a true digital zoom? Till around 3-5x im pretty sure, its just narrows the area of the sampling, but can you go up to 10x without true digital zoom? Someone should do the math, to see, what is the zoom setting, when you use exactly a 1920x1080 area. I don't think that its over 10x, so around the 8-10x settings maybe its a true digital zoom (when the cameras just blow up your images, for example every cellphone uses this method, but there bunch of other gadgets what do the same)