I'm certainly not a qualified speaker designer, but the difference in arrival times has got to be so short, in the range of a few cycles even at 15KHz-20KHz, I can't believe this is an audible effect. Have you run experiments between coherent and non-coherent alternatives and heard a difference?
Actually yes.
About 10 years ago, I was involved in a research project with my son over instrumental synthesis to avoid sound sampling. The problem with building music from sound sampling is that it precludes altering the dynamic in a musical way.
This is a long story, but essentially I recorded some live instruments in a time and phase coherent fashion. This included violins from Italian masters.
Then my son mathematically sorted out the the harmonic structure of the sounds with carefully written programs.
Now certain things came out as expected. The most complex harmonic structure belongs to the human voice and the violin. It is no accident that composers and musicians have regarded the violin as most akin to the human voice.
I and others have long known that it is fairly easy for a reproducer to do a good job of reproducing a French horn. It harmonic structure is not that complex.
The violin and the human voice are the very devil for reproduction systems to get right, or even passable for most.
Now we spent a lot of time focusing on the violin and its the relationship of its harmonic structure to the fundamental.
To cut a long story short, we found that Ted Jordan was in essence correct that maintaining all the correct relationships of the harmonics to the fundamental was vital to capturing and being able to even making any violin sound synthesis.
It is no accident that the violin has a very hard time making friends through recordings, but not in the recital hall. Listening to the the Bach violin sonatas on all but a very select few reproducers is an excruciating experience to many.
This project convinced me that this has a lot to do with the wave envelope and preserving it intact as much as possible is important.
As far as general audibility is concerned that is problematic because as I have pointed out previously, most commercial recordings are awash in violence to time and phase and how much further damage the speaker can do is mute.
However I have to stress there are bigger fish to fry in the total concept of a speaker than time and phase coherence. In fact you can do a lot of damage trying to achieve it.
However my experience with full range drivers is actually very revealing. I set up a demonstration with a full range driver system for the engineers at MISCO some months ago, and they were in many ways astonished. I did use carefully selected recordings. Obviously there are significant limitations to full range drivers, but they are revealing of the significant trespasses of crossovers.
In my designs I try and let acoustic roll offs do as much of the work as possible, with the minimum required from electrical crossovers commensurate with very smooth frequency response, lobing pattern and dispersion control.
To achieve the above goals using only first order crossovers is in my view an impossible task.
And by the way, it is a myth that first order crossovers have no time and phase shift. They do and its 90 degrees and a quarter cycle.