That's the reason why they are commonly called Bettelampeln ("begging lights") here in Germany, as you have to basically beg to even be considered as traffic participant sometime during the next cycle.

This is usually also tied to timing, just not in the sense people hammering the button to cross the street faster want. Without pushing the button, the lights will change based on car availability. Pushing the button often won't get the light to change sooner, but it will likely stay longer to allow for people to cross the street.

It may have done that regardless due to traffic patterns, but the button guarantees it has a minimum safe crossing time.

Yeah, I have forgotten to.press and never get the walk sign.

Also, I have been driving through the university of St Louis area a lot recently and there is a crosswalk there that is just for pedestrians crossing, with a light. You press it, and the light turns red stopping traffic so everyome can cross.