Those buttons that are supposed to give pedestrians time to cross the street safely have been a mystery my whole life. Up until this point, I have heard no better explanation of their use than conspiracy theories. Mainly, the buttons are a placebo to keep people standing around patiently, or give them something to do. This video seemingly ends the discussion:
This detailed walkthrough of the wiring and logic behind pedestrian buttons is fantastic. The buttons do, in fact, do something, but they are not there to help you wait for less time, merely to signal that you exist.
Next time you repeatedly slam that button, remember that the only push that matters is the first. I wonder if that is the same thing for elevator door close buttons.
For some of the walk buttons, at least, that's why pressing the button triggers a synthesized voice that says, "wait".
Personally I'm fascinated (mildly) by the variations in button style, from the big honking domes down to tiny recessed buttons that you can pretty much only reach with a bare finger.
At the crossings where I live, only if you press the button will it turn green during the proper phase. Otherwise it will stay red and let the cars turning right just blast through.
There are other buttons, that trigger audio signals for the blind, but those are not as common.