In regards to d500 vs full frame (sony vs canon): The d500 does pretty well with low-light focusing so long as you have a fast lens. The d750 is even better, maybe Nikon's best low-light camera (I'm not sure how the d5 compares, but that's way more expensive).
I would say you want an f/2.8 or f/4 lens. f/4 = x2 light of f/5.6, and f/2.8 = 4x light. Nikon and Canon will focus at fastest f/stop and then stop down (in most cases). Most sony's focus at the set f/stop, so they are not low light champs if you want to stop down. The A9 and A7RIV have a focus priority mode that will focus wide open. I'm not a sony expert, so maybe there's a way others can be used for low-light stopped down. Though if you shoot wide open, it doesn't matter. Of course, different lenses will have different issues with focus shift between f/stops, so your results may vary.
Low light focus performance and high ISO performance usually improve with lager sensor pixels, so an A7III (not R) or A9 or d750 or d4/d5 do well or similar in the Canon line up. But I suspect it's more of a lens speed issue than a low light focusing issue with the d500.
The Tokina 11-20mm f/2.8 DX is a great lens on the d500. They also have an older 11-16/2.8 that is great, but the 11-20 is newer. I had the original 11-16 (model I) back with the d7000 (or d7200 I don't remember any more) and really liked it.
Nikon has a lot of f/2.8 and f/4 glass in most every focal range, so you could find it if you look for it. The Sigma art line is also very good from what I hear, but I have not used them.
If you can find good DX lenses in 2.8-4 they will be lighter and smaller than the full frame.
I agree about the Tammy 150-600 v2. It is great when there's enough light, but then its performance falls off pretty fast due to focus hunting. That's life with an f/6.3. I don't think it's an issue with only the tammy. No f/6.3 lens is going to be great in lower light.
Personally, I would think the 50mm f/1.8 or f/1.4 or a 24-70/2.8 or 70-200/2.8 would be the ticket, depending on your distance and the size of the flock. I use the 70-200/2.8 on the d850 (and d500 when I had it) and it really is a champion of a lens for sports and low light shooting in that focal range. It stays with super fast focus even in poor light. The 70-200/2.8 is quite a bit better than the 70-200/4 for low light, but it is noticeably heavier.
If you want to save on your gym membership, the 400/2.8 D or G lenses can be had for $3000 - $5500 on auction. As the D lenses don't work on Z mount with AF, I'd stick with at least a G if you plan on going that route in the future.
Marc