More and more often, they are "beg buttons" which must be pressed, or a pedestrian green signal will never be given.

I think it probably depends on the city and maybe even the intersection. Here, when they're working (not a given) you get a walk signal (whereas if you don't push the button it stays a solid do not cross hand). Additionally, the length of the green light is potentially extended to allow enough time for pedestrians to cross.

There's even one intersection here on a road with designated and flimsy bollard separated bike route where I can be approaching on my bike in the bike lane and it notices me and turns the light green for me which is super cool.

It most certainly changes on some roads, usually to adjust timing for safe crossing. There's a street here that's notoriously wide for stupid reasons (parking in the median lol) and if pedestrians are around it gives them about 60 seconds. No button? Usually 15, but depends a little on time of day.

There's an interesting other case I've heard about from my Mom. There was a light that, at 3am, generally never changed unless a button was pushed. My mom had to get out of her car to push it every night after her night shift. I'm not sure that is still the case, but I suppose if your car never activates a sensor (or you're on a bike or motorcycle), it makes sense.

It certainly doesn't cause the lights for the cars to change at the intersection at the corner of my apartment. It rarely ever even gives the pedestrian the little dude on the screen instead of the big orange hand.

Might tell the electronics that there's a person waiting so it'll do the audible chirps or pre-recorded verbal messages when the lights change. Haven't paid enough attention to notice whether they still happen with no button-press.