The wailing wall is not part of the old temple.its where the Roman’s had their fort.
See this link: https://popular-archaeology.com/article/wailing-at-the-wrong-wall/
“ Sam’s arguments have been built on the foundations of research by earlier scholars, most notably that of Dr. Ernest L. Martin, whose research and subsequently published book, The Temples that Jerusalem Forgot, also argued that Fortress Antonia actually encompassed the area traditionally ascribed to the Temple Mount and that the temple was more accurately located on the Ophel mound over the area of the Gihon Spring, in the old City of David to the immediate southeast of the traditional Temple Mount precinct.”
That makes more sense as the temple having a natural source of water ties in with the description of The Holy Spirit as living water.
Besides, over the centuries, a lot of sacrifices on the temple grounds would lead to a lot of blood and you just don’t see the bloodstains around the Dome of of the rock area that is currently touted as the temple site.
Butchering a sheep also requires water to clean up and using the spring of Gihon makes more sense than hauling water uphill to a barren spot.
With that, the wailing wall is not part of the second temple and what you see is the final gathering of Israel, never more to be dispersed.
The Jews go there to pray, so they think it is significant.