I have a time alignment system in my car that allows me to make minute adjustments between woofer, midrange and tweeter.
this function does make a difference in sound quality but it is more of an advantage aligning left to right speakers in a car interior setting.
time aligning the individual drivers of one speaker does have some benefit but you may not find big gains in sound quality.
the best thing it can do for you is to align the speakers at their crossover frequency
which avoids out of phase cancellations of frequencies and adds coherence and more of a point source focus of the sound image. this will lead to a flatter frequency response and better imaging. .
you will need test tones and pink noise with measurement tools to get the most out of it. no way you can adjust by ear with music.