I think your biggest obstacle here is the distance and maybe the light.
I don’t think fine tuning focus would do any good with these shots. The DoF at that distance is more than enough to cover minor focus errors. If you are able to get well focused shots at close distances I would not worry about focus issues.
The light looks kind of harsh and that combined with not having many pixels on your subject cause you to lose a lot of detail. The 70-200 with a TC20E III should preform about as well as an 80-400 so it should be pretty sharp (not as sharp as the lens without a TC) as long as your subject fills a decent area of the frame.
Even with my 400/f2.8 I would not expect much out of the coyote shot at that distance and I would not have even fired the shutter on the hawk (way to far away) unless I was trying to get a shot for identification.
The 300/f4 is an awesome lens and light weight. It is probably sharper (with a 1.4TC attached) than the 70-200 + 2xTC combo that you are using. AF could get touchy with the 1.7 and 2.0 TSs.
The 300/f2.8 is about as good as any of the big primes and works very well with the TCs. If you really want a prime and can’t find anything longer that is in your price range this is a great lens.
The 200-400 is probably worth looking at. It is on par with the 70-200 for IQ and works well with at least the 1.4TC. A lower cost alternative is the sigma 120-300/f/2.8 OS with a sigma 1.4TC (170-420/f4.)
Anther Sigma to consider is the 500/f4.5. Nikonian Richard Dubiel (Dubes) used this lens extensively for a while. He has posed a lot of stunning images in our wildlife forum.