With the 60D there are a couple 'problems' you're encountering.
One is the high ISO performance. The second is the camera's reaction speed and AF capability.
The cheapest solution might be to stay with the 60D, shoot RAW instead of jpeg files and run those through Adobe Lightroom's Denoise function. This will cost you maybe around a couple hundred a year for the subscription. It might help get the images to an acceptable point but it won't address the camera's other performance shortcomings that you've encountered.
While something like a 5DIII would probably be a high ISO improvement, and somewhat of an AF improvement, for likely a similar price you should be able to find a used 1DX with better AF performance, responsiveness, and frame rate. Just make sure it comes with a battery and charger.
In some respects, the above will just be side-grades.
For substantially better AF performance/consistency, mirrorless is the way to go. At the lower end, it would be the R10 or R7 with APS-C crop. In full frame, the R8 is basically the same 'guts' as the R6II (processor and sensor). Otherwise, a used R6, R6II or even R5, depending on how much you want to spend.
Check out the Canon USA store for refurbished options and I think there might even be a loyalty discount if moving from an old Canon camera. There might be some discussions about this on this board, or just give them a call to determine what qualifies for the discount.
Whatever you decide, shooting RAW and running the files through an AI noise reduction process, such as in Adobe Lightroom, should still be part of the process, if you want clean files.
As Ron states above..I only shoot raw and process in PsCC. I agree with his advice. I use Nik Dfine for noise BUT I am finding that ACR does a fine job on noise.
Dan
The gyms I've shot in (football is pretty close to over, here) youd be fine using up to 6400 and a 2.8 zoom. Football stadium too.
50 1.8 is probably the cheapest to get photos in low light. I've used it at iso 1600, even, on older bodies in gyms. Maybe not the greatest but you can get something especially if youre going up to 6400
great camera for sports and wildlife too