I didn't read all the post, so maybe my post will be irrelevant.
But normally, if you want maximum performance, you need to stop the lens at least
1 stop from wide open. So at 5.6/300mm it should be good. If you use a 1.4x, it's f8.
and with the 2x, it's normally 2 stop from wide open. So f11+ if you want very good
images from a lens.
Sure my 70-200 is very usable at 2.8, but sharpness is better at f4. So when I put my 1,4x on it, I use the lens at atleast f5.6+. I prefer stopping down the lens a little and bumping ISO up to 800. I find the image sharper this way. So in your original picture,
you have a too slow shutter. ( try to keep it around 1/500), you can bump the iso to 400
and stop the lens to f8.
Longtele lens are not used the same way as a normal zoom. Before sending everything
back, use it in real world condition, keep your shutter high, brace yourself... Anyway, you got the idea...