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The autofocus drive speed is how fast the lens adjusts its focus system while searching for correct focus. That speed is a factor on how fast a given subject can be pulled in to focus, and also potentially limits the fastest subject that can be tracked if it is moving toward or away from you.
The combined system of the lens and the camera, along with subject contrast and light level, are what determine over all autofocus performance, and performance includes if it works at all on a given subject, how often it works, how accuratly it works, how low a light level it can work in, and how well it tracks. From the lens side it is the drive speed, as well as the maximum aperture available, that affect performance. There could be other factors like a better microprocessor in newer lenses that adjust things "better". From the camera side, the particular autofocus system and sensor array in the camera and which autofocus sensor point is in use are major factors. As you add extenders, the optical aperture of the lens and extender gets smaller, and that means that any given focus point will have a harder time determining critical focus. That is why the extender tells the lens to slow down its drive speed, so that the camera and its sensor have a better chance of finding focus or tracking focus without getting lost. The amount that Canon picks to slow things down is undoubtedly based on averages, and also likely is on the conservative side, but it is what it is. The exact reasons why a series III extender and a series II telephoto may have a some what higher drive speed or track things better has not been explained in detail by Canon as far as I know. And of course Canon's selection of what aperture to use to turn off autofocus all together is another parameter they likely picked based on some average situations, and also to be conservative. And the marketing department may also have stuck their finger in on that decision. I really wish they would allow the user to have a custom function to turn on that would allow the system to try to focus at another stop smaller, and would love to see a firmware update for my 1DX and for all those 5D3's out there to give us all that option, of course to be used "at our own risk of it not working". But getting Canon to do a change like that is difficult.
The net effect of comparing a 300 2.8 with a 2X to a real 600 f 2.8 is that the 600 is faster and better at auto focus and focus tracking, but that the 300 with the 2X is still very useful, and more versatile, and weighs less, and costs less.
With Canon lenses, assuming they are not beat to death, they hold their value fairly well, so getting a used 300 f 2.8 L IS now, and then deciding later that you want to change to something else, or go back to the 100-400, will not cost all that much for the change. Since you mention long walks with long glass, have you considered at all the Canon 300 f 4 L IS, along with a 1.4 extender? You loose a stop, and loose 600 autofocus, but you save a bunch of weight and cost. Not trying to side track your process too much, but the 300 f4 is a nice lens.
I have never really liked my 100-400 at the 400 end, and just added the 70 to 300 to my collection to cover low weight situations. But my 300 f2.8 L IS is still my main long lens, and I use it a lot both hand held and on a gimbal mount. And until the series II telephoto lenses came out, it was considered by many to be one of the sharpest and best focusing Canon lenses you could buy.