In college, I lived close to my university’s campus near an intersection that would get very busy during sporting events and other occasions where there would be many people in town.

The intersection would set all the lights to red if it had detected no activity for a good while (such as at night when it was rarely used) and would also immediately change the green light for cross traffic to red if you pressed the crosswalk button so long as there wasn’t any cross traffic for a few minutes.

It was surprisingly sophisticated traffic control considering that most of the other intersections were just on simple timers.

I know for my nearby intersections, when the button is not pressed, even when the lights change, the walk signal will never illuminate.

Yeah, a lot of the time they're just there because people look for them if they're not and then they fidget, don't wait for the signal and dash into traffic.

The town I grew up in, the vast majority didn't do anything but they gently beeped, and if you pushed the button it would tell you the sign status until it completed the cycle. Helpful for vision impaired people.
There were a few that shifted the traffic lights to a different cycle and otherwise it never did a pedestrian cycle.

There was one though, where pushing the button nearly immediately started a pedestrian cycle.
There was a wide block with an apartment building in the middle on one side, and a grocery store directly across from it. Not long enough that people complained, but enough so that people still trotted across the otherwise busy street a lot. After trying nearly everything else they eventually put a traffic light in the middle of the block that was nearly entirely on demand for the crosswalk button, because almost any delay and the (college student) residents would just dash across.

Watching confused drivers was fun.