Maui5150, we may be wrestling with how to apply our personal experiences with our perception of the OP's need.
I see the OP's sports need to be minimal now, maybe picking up as his son grows up. I also perceive his gear acquisition as a longer path than this single purchase.
If the sports in question are outdoor where light isn't an issue, then a 7D2 with a 70-200 f4 would be the way to go -- for today. down the road, an ideal setup would be the 7D2 with the 70-200 f2.8L II. Or, if sports never becomes a priority, maybe a 6D with the 70-200 f2.8L II. But, what path to take to get there?
The 7D2 offers maybe a stop of the light advantage over the T3i with respect to noise (for RAW images). It offers more focus points, but for getting shots of a specific athlete, you will still be using a single point for focusing. The advantage that the 7D2 offers is that you can add a bunch of expansion points around that single point and you can move this group around the frame to improve composition. Additionally, the 7D2 offers 10 FPS vs. 3.7 FPS of the T3i. Finally, the 7D2 offers a burst rate of 24 RAW files over 6 RAW files of the T3i.
When indoor sports is the subject, we need to look at the full impact of low light, beyond noise.
The 7D2's noise reduction may offset the loss of a stop with the 70-200 f4 vs. the f2.8 II. But, what are the ramifications of focus lock under low light with a 4.0 lens vs. 2.8 lens when using the 2.8 sensitive focus points? I suspect that the OP will see an improvement in his center-point focusing performance with the 2.8 lens over the 4.0 lens. Plus, the 2.8 Mark II lens reportedly has the best focusing performance of all the Canon 70-200 zooms. I'm not suggesting that the T3i with the 70-200 2.8 will outperform the 7D2 with the 70-200 4.0. But, in low light, the 2.8 lens will improve the performance of the T3i while the 4.0 will limit the potential of the 7D2. I have noticed focus misses with my 24-105 4.0 on both my 7D and my 5D3 that I don't experience with 2.8 lenses (17-55, 24-70, or 70-200).
Keep in mind, we are considering kids in sports. We aren't, yet, talking about a lot of speed. I see no issue with center-point focusing for these events. I've spent many years shooting center point with an XT. I also don't see the 3.7 FPS as an inhibiting factor. Burst rate has its place, but ought not be a crutch for poor timing. If anything, limiting one to a slower rate may improve one's skill set in timing the shot.
I shoot most of my sports with a 5D3 and still use the 7D for outdoor events. There are times when I use the burst mode, but I think it can be over-rated. Again, this depends upon the sport and the level of play. In my experience, even the 8 FPS can be too slow to be a reliable alternative to timing the shot. For example, try shooting a kid throwing a free throw in basketball and getting that shot where the ball is only inches away from the hands after the throw. If you start bursting as the shooter begins the throw, you will miss it. You'll get a shot before the throw followed by one with the ball leaving the frame. A lot can happen in 1/8 of a second. The best chance of getting this shot is by improving your timing. Of course, the 10 FPS of the 7D2 or the 12 FPS of the 1Dx, just might be fast enough to offset poor timing. But, it isn't hard to get this shot with a T3i.
Again, we are talking about young kid's sports here. When I shoot younger figure skaters for the club, my 60D (with a focus system similar to the T3i) does a fine job. It is with the older, faster athletes that I see a distinct benefit of the 7D and 5D3 focus systems during AI SERVO. Better athletes need better focus systems to track them. Younger athletes, not so much. I believe that the biggest hindrance that the T3i presents with kid sports would be the buffer. You do have to be conscience of it filling up. If a skater is preparing for a jump, don't fill the buffer with shots of the approach. Save some room for capturing the jump itself.
Add to this the OP's inclusion of the faster 85 (which is a fine lens) and I still think that the 70-200 f2.8L II offers more bang for the buck now. When light and fast action aren't part of the picture, the T3i sensor can challenge the performance of the 7D2 sensor. The 70-200 2.8 will give the OP greater control of DOF and a greater opportunity for fantastic portraits and candids.
I still recommend the 70D, the 7D2, or a full frame body down the road. If the fast action (which excludes the young ones) isn't an immediate need, why not wait until the price of the 7D2 drops and benefit now from the good deals on the 70-200 2.8? Then determine which body will best serve his need.
I certainly have a bias here. If the OP really gets into photography, I do think the 70-200 f2.8L II will at some point have a place in his bag. I'm just not sure which body is best. I can't advise getting lenses that would become obsolete by the 2.8 zoom. Further, it would be a shame to make a cumulative investment in lenses now that would inhibit a future purchase of this 2.8 zoom. I think this lens is THAT good and will help him define his direction in photography. It did with me. It was a big help to my then immediate need in low light sports with a crop body. Now it's an even better lens on my 5D3 -- together better in low light, more useful focal range, greater DOF control, and sharper on FF than crop.
Ultimately, I don't know if my assumptions and perceptions accurately reflect the OP's current interest and need. But, I do hope that offering the differing experiences and views in this thread will help him weed out those that don't apply and come to a conclusion that best serve's his current need and his future acquisition of gear.