Wayne
Excuse my ignorance but what is the DX mode advantage you are referring to?
My statement may not have been the most clear possible.
If we put a 50mm lens on a FX camera, then it sees the wider field of view width we call 50mm equivalent (equivalent to FX).
If we put same 50mm lens on a DX camera, due to the 1.5x crop factor, then it sees a cropped (more narrow) field of view we call 50 x 1.5 = 75 mm equivalent (DX with 50mm lens is equivalent of field of view of a 75 mm lens on FX).
It is still the same lens, still obviously 50mm, but with the same lens, the cropped DX sensor sees a smaller area, like a FX sensor would see with a 75 mm lens (called equivalent, meaning field of view).
With a zoom lens, as we zoom in to a more narrow view with the lens, the flash with zoom follows it, same zoom values, to concentrate the beam into a smaller but brighter beam. The flash need not cover a wider area than the lens can see.
Some flash models (SB-700 and SB-910 for example) also have a DX mode. So basically, then when using this 50 lens on a DX camera, they can zoom the flash to 75mm (a smaller concentrated beam that is a bit stronger). That will still cover the smaller scene area that the cropped sensor sees.
So my meaning was, on a DX camera, this concentrated DX beam could "equalize" a weaker but concentrated flash (SB-700 that has DX mode) with one a bit stronger that did not do the extra DX zoom. The SB-700 is a bit weaker flash, but on a DX camera (for direct flash), it can be a bit stronger.
Not always true, for example, if the opposite... if we zoomed the DX wider to see the same FX view, then this advantage is lost. This would seem more usually the case, instead of the first case. So maybe not a valid DX/FX comparison, but the DX zoom does help the flash on a DX camera (direct flash). Specifically, it uses the FX 24mm flash zoom on the DX lens 16mm zoom setting.
I think the Yongnuo YN565EX is supposed to have the DX mode... it has the DX menu, but which does not work on my Nikon version, simply is not implemented. I am guessing perhaps the Canon version has it?