Why is it that Hashem promised never to destroy the Kosel?
The Shoel U'meishiv answers that this can be understood based on Rashi's and Tosfos' opinions in Sukkah(41a) where it says that the Third Beis Hamikdash will not be built by hand, rather it will decend from Heaven. There is a principal that the blessing does not come on to something that is empty . For example when Eliyahu Hanavi got oil for Ovdiah's wife and children he asked her first "is there anything left in the house?" and she replied "a little oil." From this oil there was blessing and it filled her and all the neighbors containers and jugs. If she would have had nothing it would not have worked as per this principal. This is also the source for people who during Bentching leave some type of crumbs or foodstuff on the table because you can not receive blessing on the food if there is none there. With this principal explained we now know why we require the Kosel to remain in order for Hashem to send down the third temple, for there has to be somthing here - it can't be empty - because a blessing does not come on to something that is empty .
The Zohar writes (2:5b, Parshas Shemos)
תשורי מראש אמנה, מאתר בי מקדשא דלעילא ומאתר בי מקדשא דלתתא, דאמר רבי יהודה, מעולם לא זזה שכינתא מכותלי דמערבא דבי מקדשא, דכתיב: 'הנה זה עומד אחר כתלנו', והוא ראש אמנה לכל עלמא
"Sing from the heights of Mt. Amanah [homiletically referring to 'heights of belief']: From the place of the high temple and the low [earthly] temple, as R. Yehudah says, forever the Shekhinah will not move from the West of the Temple, as the verse states "behold it is standing behind our wall", and this is the heights of belief for all the world"
Thus, the fact that the Western Wall remains standing is meant to be a testimony to faith, because the Shekhinah will not leave from there.
As to why the Western wall, as opposed to any other wall: the Western wall is the wall closest to the area where the "Holiest of Holy" was, the place of the Ark and the Foundation Stone (Even Hashesiyah). See his shiur from R. Levi of Yehivat Har Etzion, where he also explains the relevance of the west being associated with Shekhinah more generally, due to the proximity of the wall to the Holy of Holies and to the fact that the Temple as a whole faced west.