In regards to your thoughts on moving up to full frame, here is a pair of shots that reflect my experience going from 50D to 5D3 with 70-200 f/2.8L II when shooting outdoors sports at distances comparable to yours.
The first shot is with 50D, using f/3.2 @70 mm, 1/1000, ISO 100, using tripod and "one shot" focusing. This looks pretty average in sharpness among the kid soccer pictures that I got with this setup last year. I did manage to get a few sharp ones, but this shall be attributed to good luck rather than a great skill. I did not find Al Servo on 50D that useful ... focus switched to something else too often.
The second shot is one of the very first with 5D3, using f/3.2 @168 mm, 1/2000, ISO 640, also tripod but now "Al Servo" focusing. My rate of sharp soccer images is still not particularly high, but the ones that are nicely in focus are much better than anything I got with 50D. Beside better autofocusing, I feel more at ease in pushing up ISO, enabling faster shooting. For example, the second goalie shot was taken at 8 AM and the sun was not exactly cooperating with illuminating the subject. BTW, both all taken as single shots rather than continuous shooting.