In many languages, muting the sound and watching the lip movements is enough to guess palatalized/non-palatalized. Is there a reason for "Why palatalized consonants are commonly accompanied by stretched lips?". Does having broad lips help producing palatalized consonants, or is it just to increase "perceptual distinctiveness"? (If it were the latter, I guess there would be some languages where, instead, non-palatalized sounds are produced with broad lips??)
If the consonant in question is neither labial nor labialized, the lip position accompanying its pronunciation is characterized only by that of the neighboring vowel, it doesn't depend on whether the consonant is palatalized or not.
For example, in Russian most consonants go in pairs, non-palatalized (“hard”) vs. palatalized (“soft”): /t/ vs. /tʲ/, /d/ vs. /dʲ/, /l/ vs. /lʲ/, etc. There are five vowel phonemes in Russian (/a/, /e/, /i/, /o/, /u/), and palatalized consonants can occur before any vowel, that is, for example, /lʲa/, /lʲe/, /lʲi/, /lʲo/, /lʲu/ are all possible combinations. So when you say the word лес /lʲes/ “forest” or лис /lʲis/ “fox”, even before you actually begin pronouncing the initial consonant, your lips already accommodate, stand into the position for the vowel that will follow. Since these words have the closed /e/ and /i/, the lips do stretch, but that would also happen if the word started with the corresponding vowel. On the other hand, in the words лёд /lʲot/ “ice” or люд /lʲut/ “folk”, the vowels /o/ and /u/ are rounded, so even before saying these words your lips get rounded and protruded forward.
If you mute the sound and watch someone pronouncing лис /lʲis/ “fox” and люд /lʲut/ “folk”, you will see stretched lips in the first case and rounded protruded lips in the second case. And in the case of ляг! /lʲak/ “lie down!” you will see lips in a neutral position.
I call this phenomenon “accommodation”, you can call it “coarticulation”, in any way it is a kind of assimilation, but, at least in Russian, it is absolutely insignificant from the point of view of phonology, there is no need to speak about allophones [lʲ] vs. [lʲʷ].
Anyhow, it is the quality of the vowel, not that of the consonant that defines the lips position.