The amount of reflected sound in proportion to the direct sound should be affected very much by the dispersion. The arrival time of the reflected sound should be exactly the same if the speakers and listener is in the same position. Dispersion does not affect the speed of sound.
Dispersion does not effect the ‘speed of sound’, which is obviously a scientific constant, but it does clearly impact the available routes of the sound to the ear, and therefore the speed of post-delay the brain has to deal with. With a highly directional speaker, say a horn or ESL 57, hardly any sound in the smaller wavelengths (mid & treble) hits the side-walls, floor, ceiling etc until after it has first hit the listener’s ears and later bounced off the back wall and back into the room. Maybe easier to get this point across if we consider an omnidirectional speaker that splashes sound out in all dimensions, the time delay from direct and reflected sound becomes hugely blurred as effectively it is arriving not at one time but many different ones, and many of them are far closer to the original signal. FWIW I’m not knocking omnis, many love them, and I can understand why, though I do view them as an effect and absolutely not what is heard in the recording studio.
A wide dispersion box speaker is kind of a middle-ground allowing a lot of treble and mid energy out to the side, up, down etc which will inevitably hit the listener long before say a horn speaker or ESL located in the same position.
As ever with hi-fi it is a compromise/personal choice thing, e.g. I always set a system up for a hot-seat and don’t really care about filling a room. As such having a distinct lack of treble off-axis (as Tannoys etc do) is really no issue to me as I’m just not sitting there. As long as I’m not getting the real ‘head in a vice’ thing where a speaker is beaming so acutely you can’t move a few inches without losing top end I’m happy (57s can certainly do this, Tannoys to a lesser extent).
PS As stated above I consider LS3/5A type nearfield mini-monitors as their own thing and largely excluded as typically you are listening to them in such a tiny listening triangle of just a metre or two well away from any walls. Again you get the direct sound far further ahead of the room splash, so same effect regardless of dispersion, though I’d not enjoy them in a room without a good thick carpet. I am increasingly convinced this is why my favourite speakers are either horns, panels or mini-monitors, and also why I hate typical hi-fi speakers in typically sparsely furnished rooms (my rooms are deliberately very well damped, I do not like adding random reverb to a recording with the room!).