As for wanting constant f/5.6, I don't see that as important or even useful. Remember, the camera body tells the lens what aperture to use and the lens has to figure out how to do it.
Actually, it is a big deal, isn't it? Wouldn't a constant F/5.6 max aperture still allow autofocusing with a 1.4x teleconverter on the right Canon bodies?
(Forgive me if I'm off here -- I never shoot with long lenses like these.)
There are two issues here: (1) the maximum aperture being constant throughout the zoom range and (2) that maximum aperture anywhere in the zoom range being wider (i.e., numerically smaller) than f/5.6.
Regarding (1), the Canon 100-400 L, which I own and use extensively, has a maximum aperture which varies from f/4.5 at 100mm to f/5.6 at 400mm. When I use the camera to set the aperture to any value between f/5.6 and f/32, that value is held regardless of the focal length that I zoom to. If I set the aperture to f/4.5 at 100mm, then zoom to 400mm, the aperture automatically changes to f/5.6. Making that lens a constant maximum aperture means that maximum aperture would have to be f/5.6. Why should I give up 2/3 of a stop at 100?
The effect is even more pronounced with the 28-200 Canon lens. At 28mm, the maximum aperture is f/3.5, at 200mm, it is f/5.6. Who wants to use a 28 f/5.6 lens?
Regarding (2), somehow third party lens manufacturers get their lenses to lie to the camera body. The actual maximum aperture at 600 is f/6.3 but the lens tells the body that it is f/5.6. Since the difference is only 1/3 stop, I suppose the image is underexposed by that amount, although I've never tested it. So why don't Sigma and Tamron make 150-600 f/5.6 lenses?? My guess is cost and market positioning. To maintain the same 95mm aperture diameter, the maximum focal length of these lenses would only be 532mm. I suppose both manufacturers thought fewer people would buy, for example, a 130-520 f/4.5-5.6 or a 150-500 f/5-5.6 that weighed and cost 50% more due to its 107mm front element.
Hope that helps.
Correction: "... or a 150-500 f/5-5.6 that weighed and cost..." should be "...or a 150-600 f/4.5-5.6 that weighed and cost... "