Is there a special status awarded to the plants (shrubs?) growing out of the Kotel? Why don't they get rid of them, as their roots obviously have deleterious effects on the wall?
The Mishna on Megilla 28a rules that a destroyed synagogue retains holiness, and if grasses grew on it they should not be picked because they add to the feeling of despair. (The subsequent Gemara on 29b discusses picking the grasses and leaving them there, though the Rambam (Perush HaMishna 3:4) and the Mishna Berura (OC 151 sk 29) both understand this to be optional.)
Though the Kotel is not a destroyed anything, I think similar logic may apply.
The relevant part of the Mishna:
ועוד א"ר יהודה בית הכנסת שחרב אין מספידין בתוכו ואין מפשילין בתוכו חבלים ואין פורשין לתוכו מצודות ואין שוטחין על גגו פירות ואין עושין אותו קפנדריא שנאמר והשמותי את מקדשכם קדושתן אף כשהן שוממין עלו בו עשבים לא יתלוש מפני עגמת נפש:
And Rabbi Yehuda also said: a synagogue which was destroyed we do not [do certain mundane activities in it] as it says (Vayikra 26:31): "and will bring your sanctuaries unto desolation". We see that even after they are destroyed, they are still sanctuaries. If grasses grew in them, one should not pick them because of [inciting] grief.
It could be that pulling them out would cause direct damage to the wall, while leaving them there is only indirectly causing damage, and that we prefer the former.