Well, moving to a 7D2 you gain some FPS and robust AF and just about barely 1 stop of ISO performance. But, you're still going to be limited by the focal length and focal-ratio (speed) of your 55-250 and the FOV it produces. So little will change in that regard. Distant action is a lot easier to nail, so you may not notice much of anything if you're objective about it. Close range action can be more challenging to the AF system as depth of field crushes down more and you may notice more keepers there with a newer body.
Overall, having more reach on the field gives you more options. A 100-400 is a great thing to have as the field gets bigger. The AF speed of that 100-400 is also a big plus. But, I would actually point you more towards perhaps a Sigma 120-300 F2.8 OS if you're wanting a sports lens with some versatility. Constant F2.8 speed is a big help to keep the shutter speed fast as the light drops. And you can put a 1.4x TC on it and keep really great performance, having it at 168~420 F4 OS at that point, which is better than the 100-400 flavors. It's a big lens, but the F2.8 OS (non-sport) model is not much more expensive 2nd hand than the 100-400's and a heck of a lot better for sports action and will be more versatile. Heck, it can go to a 600mm with a 2x TC even if the light is really good. I'd do this over a new body first.
After that, yea, a new body could help, such as an 80D or 7D2. Used/Refurb 80D's are really affordable too. Or you could shop a used 1DIV. On the cheap, grab a used 1DIII.