Give and take. At the lower focal length there will be less DOF blur, but you have to enlarge it so what the shot gives you the enlargement taketh away. And that is different from movement blur. Once captured, movement blur is locked in.
The original question posed by Dreed, as I understand it, asked if the unavoidable f/stop drop resulting from the use of a teleconverter changed the effective DoF of the resulting image as well as the effective focal length, or was such a drop in F/stop merely equivalent to a drop in T/stop (transmission loss).
After some initial confusion on this issue, and with the help of the authoritative Bart, it was decided many posts ago that there is effectively no change in DoF as a result of the use of a teleconverter. In other words, the drop in F/stop corresponds with the increase in focal length to maintain the same DoF, broadly speaking.
Now, when comparing the technical qualities of two images, whether such qualities be resolution, detail, noise, DoF, or even perspective, it is essential to compare equal size prints, or equal size monitor images, of the same captured scene viewed from the same distance.
If one doesn't do this, then one can achieve almost any result that one's biases lead one to. To take an extreme example, one could shoot a high-resolution image with a telephoto lens at F2.8, producing a very noticeable shallowness of DoF on a large print, then reduce the print size to that of a postage stamp and claim, "Who said that wide apertures result in a shallow DoF?"
One reason shooting with a TC is preferable is framing and focusing. Just easier to get and keep the focus spot on the target. In addition, metering is better.
In my opinion these would be the main advantages of using a teleconverter, which are similar to the advantages of using any telephoto lens of equal focal length.
One the other side of the coin, let's look at the advantages of using a 70-200/F2.8
without 2x teleconverter, shooting the same scenes.
First, one gains that very significant advantage of the rangefinder camera where the scene one sees through the viewfinder is larger than the shot one intends to capture. One is thus able to anticipate events more easily, as they unfold, which should enable one to do a better job at 'capturing the moment', which is surely a very important factor when photographing sports. Henri Cartier-Bresson used this quality of the rangefinder camera to great effect.
Secondly, that 2-stop advantage in noise is always there, when using the lens without 2x converter. The issue of a possible requirement for a faster shutter speed due to the longer focal length, makes that a 3-stop advantage. The noise differences between ISO 100 and ISO 800 are very significant for me.
Thirdly, I would think that anyone who buys a rather expensive and rather heavy lens such as the 70-200/F2.8 does so because image quality, sharpness and resolution are important to him. Is it not contradictory that such a person would be satisfied with effectively a lower quality lens of double the focal length?
Fourthly, there is no weight-saving when using a 70-200/F2.8 with 2x converter instead of the new Nikkor AF-S 80-400 G. In fact, after taking the trouble to look up the weights on the internet, I see the 70-200 with converter is about 300gms heavier.
I rest my case.