That seems unlikely to me. If they were fully timer-based, that would mean there are times when cars would have to stop even though there are no pedestrians trying to cross, and we can't have that(!)

in NZ they are on every intersection, you don't get a pedestrian crossing signal without someone pressing it. Even better are the diagonal cross intersections where all the cars stop once a cycle for pedestrians to cross to whatever side they want

maybe in america, definitely not everywhere else. In sweden pressing the button can sometimes give you instant green depending on the specific signal and if there are any cars at the intersection.