Call me a contrarian, but it is not THAT much life. Sure the 70-200 f/2.8 IS II will work fine on a t3i, but not exactly the best focusing. Better off going F/4 at that point and saving cash. 70-200 f/2.8 IS II on a t3i is like Snow Tires on a Ferrari... Sure it works, may be better than nothing, but sort of missing the point.
I'm relating to personal experience.
I am relating to personal experience as well. I had a t2i and at that time owned both a 70-200 F/4 (both IS and non-IS) and used an 70-200 F/2.8 IS II which I later got down the line AFTER I got my 5D MK III.
You can shoot sports with ANY camera, better glass helps, but... for sake of comparison...
a T3i with a 70-200 F/2.8 IS II vs a 7D MK II with a 70-200 F/4 IS (heck maybe even non-IS)
In short you lose a stop on the glass but gain not only 64 more cross-type AF points, but far more sophisticated tracking.
I noticed a huge difference in the 5D MK II vs the 5D MK III for shooting sports (specifically 30+ MPH cyclist for example) and drool at what the 7D MK II appears to be capable off.
For what ever you give up on glass speed, you get better ISO performance with the 7D MK II... not to mention almost TRIPLE the FPS.
You want 2.8 and IS, you can get into the Sigma, which is a step down for sure from the Canon, but the Sigma 70-200 on a 7D MK II vs a 70-200 F/2.8 IS II on a t3i, I am betting on the 7D MK II to capture more in focus and have a much higher keeper rate, and for not that much more.
While the 70-200 F/2.8 IS II can "bring out the best" in any body, there is not that high of a bar on a t3i.
And you want to capture sports?
65 cross type vs 1 - Which will nail the focus?
Which camera will track better?
Which camera has better ISO / Noise? I would give the 7D MK II 2 stops on the t3i
You are shooting action in a gym... with the 7D MK II you can fire off a burst of 10 FPS vs 3.7 FPS - Any bets which has a high chance of getting keepers or THE shot?
Better sensor. Better AF (tracking). 3X FPS. Better IQ. 65-Cross-Type AF points vs 1?
? Better High ISO
I love my 5D MK III and I love my 70-200 F/2.8 IS II. It is probably my most used config followed by 24-70 F/2.8 II... But it is not a sports combo. I shoot sports with it, but it does miss.
Just an opinion but the 7D MK II with moderate glass even an F/4 or not IS is going to get more images and more keepers than the t3i.