It's often better for pedestrians when there's a button. Because the light usually changes pretty quickly after pressing the button. Otherwise you'd have to wait until the timer switches the lights.

It's usually used for pedestrian crossings where pedestrians are rather rare.

Intersections where cars have to pass detectors (induction coils in the ground) to get a green light exist. Same principle: only switch the lights if there is demand for it, with a default oriented at the most likely demand.

The pedestrian-default at least around here is implemented by making entire zones slow-driving and pedestrian-priority, not on single intersections. If cars are allowed higher speeds, it IMHO makes sense to have pedestrians push the button and wait a few seconds, since they can do that easier and more efficient.

I'm sure there's many answers, but one would be that it's hard to determine a pedestrian's intent to cross. A vehicle at an intersection, in the right lane - yep, you can be fairly sure its intent is to cross. A pedestrian hanging around one end of a zebra crossing can be waiting for a green light, grabbing a smoke, chatting to a friend or playing Pokémon Go for all you know. There's no good way of automatically knowing that they intend to cross.

Because some intersections are programmed to maximize vehicle traffic throughput, and only allow pedestrian crossing in response to pushing the button.