I believe the Halacha is that you're supposed to tear your clothes when seeing the Wailing Wall, just as one does when 'sitting Shiva'. Why don't most people?
There are quite a few ways in which people avoid tearing, though I guess most don't do it simply because they are unaware of the Halacha.
The Kotel is not the best place to tear - one should try and see the makom hamikdash (Temple Mount).
Unfortunately, the vast majority of Jews who visit the Wall are not (as yet) religious. Most of them don't know the rule that one should tear his clothes and, if they did, most would most likely not abide by it, so as not to ruin their clothes for the sake of a rule they don't consider obligatory.
(Among religious Jews who visit, many also don't know the rule; others have visited recently enough that there is no obligation to tear clothes. See also Zvi's answer.)
 Citation needed, but I think it's pretty obvious from looking around.