I have run in to a lot of 4-way intersections where the button doesn't change the timing, but if the button was never pressed, once the traffic lights change the walk sign will not come on.
It needs to have been pressed before the light change to show the walk sign. It's either dumb, or it really did increase the delay until the next light change, to allow a pedestrian to cross. It just didn't speed up the initial light change.
This sounds like a reasonable compromise if you optimize for vehicles.
Also it means you can possibly have shorter intervals between pedestrian crossing openings because they won't affect anyone as long as nobody pushes the button.
On 3rd street in SF (effectively 6 lane road with 2 lane roads intersecting) they have this setup at most intersections. It makes sense for crossing the 6 lane road, especially since it makes the crossing light extra long to allow for crossing the wide road in time. There's not too much cross traffic, so no point in slowing the 6 lane traffic down unless there's someone crossing.
But they also have the no cross light until the button is pressed for the 2 lane crossing and I think it actually makes things more dangerous. If you don't hit the button early in the cycle, you of course have to wait until next cycle.. Which can be a while! Not that many drivers are turning.. So most people in the neighborhood jaywalk. Which means that drivers looking to turn see a red hand and probably think it's okay to turn even if it's not.
Except it trains drivers turning left through the intersection that pedestrians do not have the right of way until they suddenly do. The intersection at my apartment does this and I nearly get hit about once a week by some idiot turning left who assumes I don't have the right-of-way.
It means that if a pedestrian arrives just as the light is changing, they have to wait through an entire cycle for a walk signal. I just dealt with this a few minutes ago. Annoying.
It needs to have been pressed before the light change to show the walk sign. It's either dumb, or it really did increase the delay until the next light change, to allow a pedestrian to cross. It just didn't speed up the initial light change.