Why do crosswalks still have the button on the pole to “push” when you need to cross? Aren’t all of the lights timed anyways and will change no matter if you push the button or not?
Here’s what The Master says.
There are many intersections in my area that won’t change at all unless someone pushes the button, or a car comes along and activates the sensor. There are others which are on a timer, but the cycle does not include a “pedestrian walk” unless someone pushes the button.
No, sometimes (like at intersections which get very little cross-traffic coming from one direction, and lots going in the perpendicular direction) the light will only stay green for a couple of seconds (not long enough for a person to cross) unless someone presses the crosswalk button.
A curious phenomenon in my part of L.A., is that I frequently see people pressing the button maniacally, as if pressing it more is going to make the light change faster. I haven’t conducted a controlled experiment for this, but I somehow have my doubts. 
And as Cece alluded (although it may not have been clear, since his test light was only for pedestrians rather than at a street corner), while the lights for the cars may be timed to allow transit in all directions, the specific “Walk” lights may remain off when no pedestrian “asks” for the service.
At an intersection near an office where I worked, the lights for vehicle traffic would cycle through for both directions, hour after hour, but the “DON’T WALK” signs remained lit until someone pushed a button, after which the “WALK” light would flare (briefly) at the next change of lights for the cars going the same direction.
There’s also the style of crosswalk that’s not connected to the stop lights at all. It’s just a pedestrian sign and a light, you press the button and the light starts to blink, the cars must stop while the light is blinking. The button is the only deciding factor on whether the light’s on, so it’s rather necessary.
We’ve just got them here a year or two ago, but I’m sure they’re not new.
If you follow them to their office, you’ll see the same people poking the elevator buttons seven times.
Some of the newer crosswalk buttons in SF beep when you press them. No idea if the audible feedback actually results in people not flailing away at the buttons as much.