In my opinion, this controversy comes from a misunderstanding of wave behavior. Sound can behave as particles or waves depending on the wavelength and the distances between reflective boundaries (ie. room walls, ceiling, and floor). Most of us can intuitively imagine how particles bounce off boundaries, as if they were ping pong balls. But waves aren't so easy to characterize. Waves of short enough wavelength, higher frequency, for all intents and purposes act as if they were ping pong balls.
As the wavelength gets longer, approaching the size of the speaker cabinet width, or the dimensions of a room, much different things happen. Waves can propagate, bend, reflect and diffract in non-intuitive ways. And wave phase begins to matter greatly. We don't have to think of phase angle when ping pong balls bounce.
When wavelengths are in the same order of magnitude as room dimensions there is a rough dividing point, called the Schraeder Frequency. In many rooms in our homes, the Schraeder Frequency is roughly 250 Hz. Waves with shorter wavelength act more like ping pong balls, and wave with longer wavelength behave in more difficult ways to understand, especially if you prefer to imagine them as acting like ping pong balls. Below the Schraeder Frequency, waves reflected off of room boundaries have a major effect on sound. Above it, much less so. The controlled dispersion idea seems to assume that all wavelengths behave as do the waves longer than a rooms Schraeder Frequency. Maybe it helps to think of room dimensions in terms of wavelengths and not feet. Sound travels at 1126 feet/second. Sound at 100 Hz (100 cycles/second) has a wavelength of 1126/100 = 11.26 feet. So a room with walls 22 feet apart are 2 wavelengths apart for sound at 100 Hz. For other wavelengths… you do the math.
I think the controlled dispersion idea may have some merit for frequencies below a room's Schraeder Frequency. Above that, not so much. But most or all of the horn speakers claiming controlled dispersion benefits, control the dispersion of the tweeter and not the bass driver.
The proper way to increase the dispersion of a woofer is to lower it's crossover frequency. If you run it too high, it's dispersion gets narrow and narrower. The lower range of a tweeter will disperse sound over a very wide angle. If a woofer is crossed over too high, there will be large and audible discrepancy between its dispersion and the tweeter's. So the controlled dispersion idea argues to limit the dispersion of the tweeter in order to avoid this dispersion discrepancy. I would rather avoid this problem by crossing over the woofer at a lower frequency. This may require a more expensive tweeter, but it sounds much better to have dispersion as wide as possible.
The effects of reflected sound at wavelengths shorter than a room's Schraeder Frequency just don't matter like they do at longer wavelengths.
Where is Floyd Toole when you really need him?