Generalizing from the situation in a busy, pedestrian-oriented city to elsewhere would also be a logical fallacy. In a place like Manhattan or downtown SF, where both heavy traffic and large numbers of pedestrians are expected, the default light cycle already attempts to give both the traffic and pedestrians a turn. But in Silicon Valley there are many crossings where you will never get the “white walking man” if you don’t hit the button, and if you just cross when the cars are stopped at a red light you won’t make it, because the red light lasts less than ten seconds.
Depending on the situation, a well-designed crosswalk button won’t necessarily cause traffic to stop quicker; it may just extend the cycle to allow pedestrians more time. But if there is only a pedestrian crossing and no side street, the cars might never get a red light if you don’t hit the button.