Shooting sports in a gym is a very different than than out on a field outside. F5.6 is very, very slow for indoor in a gym, especially with a less sensitive sensor and limited tech behind ISO. So the 100-400 won't be good in the gym because it's just too slow (F5.6) and it will require maxed out ISO and you may still just end up with slow shutter speed which doesn't freeze action. The image will fall apart at max ISO unless that gym is very, very well lit for someone that's not already very well versed in processing noise. A 70-200 at F2.8 in a gym can work, but it will likely also be at max ISO or near it, but at least you'll have those extra 2 stops of light for your shutter speed to stop motion, which at least helps a lot to not have a noisy blurry photo and instead just a noisy photo. Much easier to process that.
But, a 100-400 will be super useful outside for sports on the fields where you'll want reach.
I would argue for a 100-400 outdoor and instead of the 70-200 F2.8, maybe look at a 50mm or 85mm at F1.8 for indoor in the gym on that body.