As your main focus is sports, within your budget, my top choice would be Sony A6400 ($900). Its AF ability should be the best in that price range. You may want to check out the review on dpreview and others. If you already have good Canon glasses, you may consider getting a Sigma MC-11 adapter (occasionally on sale at $150) to use the Canon lens on Sony.
If you want to go even smaller size, micro 43 system would be quite handy.
@Frank Dernie would know more on the system. But be sure to compare its AF with Sony. If you find the m43 offers good enough, Olympus could be a very good pick and probably lower price too. Also, the crop factor of m43 give you a better range on the tele side. It can be a plus for your need.
Also try Fuji XT-30 ($900), Fuji has very good color out of the box. In the past, Fuji's AF was rather sluggish. But it has been greatly improved in the latest generation. But Fuji glasses are generally more expensive. And no 3rd party makers like Sigma and Tamron.
Like what
@PierreV suggested, Panasonic FZ 1000 II ($900) or Sony RX10 (the latest generation may be out of your budget) would be a one-stop solution with good zoom range too. You can save lots of money on lenses.
If you really want to go cheap, the previous generation Panasonic FZ 1000 can be found ard $600 brand new with 3 years USA warranty. If its AF is enough for you and you don't mind the sensor is a bit dated, it can be a good low price choice.
Finally, if you want to go super handy, Sony RX100 V or VI probably can offer fast enough AF for your need too. The camera size will allow you to carry it everyday <<< I love that idea.
For DSLRs, I'm sorry. I can't think of any good choice satisfying your need within the budget.