Does Time alignment and Phase coherency make for a better loudspeaker?

Absolute phase gets thrown out the window with multi-track pop/rock recordings. 

Not being time and phase correct does not eliminate a speaker for me. A time and phase correct speaker has a "rightness", for lack of a better term, that a speaker that is not time and phase correct does not have IME. The negative that I have seen with them is that many of them will suffer dynamic compression when pushed hard. It is a trade off and depends on listening habits and what one truly values. As stated above, some either don't hear it or don't place that much value in that particular trait. To each their own.

Time Aligned, My personal Experience: YES. Size of listening space and distance effect the importance.

Absolute Phase, makes sense that it does, I just have not paid attention to Absolute Phase personally. I suspect if I did, absolute phase will produce increased distinctness. Slightly out of phase is a technique engineers use when mixing to diffuse, widen  .......

Distinctness (not brightness) is very important, it more accurately reveals what the artists and engineers released. It is the primary improvement achieved by my newly acquired CD Player

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Time-Aligned Signal PRIOR to amplification: then Maintain that with time Aligned Speakers

My conclusion while searching for a CD Player with a preferred sound(not everyone's preferred is better) is:

DAC = Salad. Combo of stable dots, then Dual Time Aligned DAC's, then the addittives of sampling, anti-jitter, re-clock, proprietary filters ... can yeild a perceptible difference, for me it is 'distinctness'

which starts out as a small difference, but when applied to everything happening, the drummer's brush work, primary singer, primary instruments, back-up singer(s), improved awareness of an instrument, imaging, each piano note. Repeat, not brightness, not glass precision, but a slight veil removed to hear the distinctness the artists and engineers intended.

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My Time Aligned Speakers

AR2x, my first 'decent' speakers when 18 years old, ollege days, listening to the Nightbird (Alison Steele) and Cousin Brucie late nearly every night,AR2x, sounded better when slanted slightly. 

JSE Infinite Slope Model 2's, Time Aligned Sloped Front, and the patented infinite slope crossover, are the most accurate speakers I own. My son has them now.

My Current Speakers, 15" woofers, horn mid and horn tweeters: preferred sound, (less measured frequency distribution accuracy than the JSE's), sound better after I added a 1-1/2" block within the base to raise the front/tilt them back to:

1. achieve time alignment (not measured), simply approximately achieved based on observations of the angle of other time aligned speakers I have seen and heard.

2. aim tweeters 'up' to direct received sound to seated ear height

3. change the angle of reflections of the larger drivers off the floor/ceiling, as well as toe-in alters the side wall reflections. I recently refined my toe-in with a sound meter, a small change yeilded 

IOW, sloped fronts for Time Alignment improvements are multiple.

Almost twelve years ago when I purchased my current set of speakers even my family could hear the difference.

"Rightness." Why not? I don’t know what to call it in audio speak. After years on the audio merry-go-round the first word out of my mouth was BINGO! It had taken me years to simply understand what was missing.

Laughingly unexpected is how quickly and easy it is to hear designs that are lacking.

Your KING OF SUBWOOFER thread sure got ’em bouncing off the walls. I'm not seeing it, did it get pulled?

Angling back the front baffle of the enclosure usually isn't enough, in and of itself, to achieve the desired results.