I apologise if you find this odd, I did not mean to offend you. Ok my telephoto zoom is the Tamron 70-300 VC, you loose F4 very quickly after 70mm for example in a low light gym at 70mm f4 iso 2500 i am only getting about 1/80, bearing in mind how heavy the 70-20 IS II lens is for me this is not a workable shutter speed for indoor sport. At 4000 about 1/125 at 6400 about 1/250 at 8000 1/320. I don't want to be shooting more than 6400. Some gyms have better light e.g. if they are part of a sports centre and the room used for other sports. No flash photography is allowed to freeze motion.
Tbh for gymnastics I’d have thought you’d have needed to be up around 1/1000 shutter speed to freeze the action, unless you’re only taking ‘posed’ photos? An f2.8 lens may allow 1/1000 with 6400 iso, although high iso isn’t a strong part of the 80D.
I’ve read all the responses and understand that you don’t want to swap gear, and I know you’ve been saving up for the 70-200mm f2.8 for some time (5 years was it?) but you basically only have a few options.
1. Your preferred option of keeping your gear and buying the 70-200mm f2.8 but then you’re just going to have to put up with the weight and the pain that comes with it.
2. But the lighter 70-200mm f4 and put up with more noise in your image.
3. Stick with what you have and put up with even more noise.
4. Bite the bullet and get a lighter system. With what you’ve saved for the 70-200mm f2.8 and what you can sell your gear for you may be able to afford the Olympus EM1-II and 40-150mm (80-300mm eq) from ‘reputable’ grey stores such as HDEW.
I understand 1 is the option you have your heart set on and 4 is your least preferred option but I’d really urge you to think long term and whether you think you’d be able to carry the 70-200 f2.8 around for long periods, even on a sling strap.
I don’t know if you’re aware but Olympus do a test and wow where you can try their equipment for free for a few days so you could try out the EM1-II and 40-150mm f2.8 to see how you get on. Olympus will feel very alien to start with but once set up to your preferences it really is a great camera.