You are right, stopping down stop may improve the image quality. The 70-200 ii i dont know, but with the 300 2.8 i nearly never had lighting conditions where i get an improvement by stopping down, as motion blur is worse than "lens softness" or "converter softness".
The 70-200 II with 2x extender just doesn't look right at F/5.6 at F/8 there is a big improvement. It's about on the sharpness level of the 400 L F/5.6 when shooting between F/5.6 and F/8.
As far as i know the 300 F/2.8 is already a very sharp lens wide open. But it's hard to actually find the reason for softness at long focal ranges as focusing problems and various vibration sources also play a role.
For me a subject in direct sunlight and the lens on a monopod with IS on and the camera (550D) on iso 400 max worked pretty well but only half of the shots are satisfying. Selecting a higher iso destroys detail and you cannot sharpen the image anymore without adding a lot of grain. So the situations where using a TC actually makes sense are pretty limited.