My city surprisingly set ones up at certain intersections, where the walk lights won’t change UNLESS you hit the button. Had to get back in the habit of pressing them again

There's one on my bike ride home that's wired backwards. You need to hit the one that controls the perpendicular flow to go straight.
I sent in a service request to the city, but they said it was the state's responsibility and forwarded it.

I had one that did nothing in my town. I wrote to the state DOT and they fixed it for me. It’s been about 10 years since then and it still works.

Yeah in my city the difference between hitting the beg button and not hitting the button is often that if you hit the button, the next time your direction of travel has the green light, the little walking man light also comes on. I'm pretty sure it doesn't affect the timings at all. And if you hit the button even a split second after your direction of travel's green light comes on? The little man does not light up, instead it makes you wait through more than an entire cycle before it gives you the little man.

New York City is awesome in this respect: They have almost no beg buttons. There is just a pedestrian phase in every light cycle, no matter what. The same way we assume a car phase in every light cycle. Because it's a city, there are pedestrians in cities, so it's assumed that they're walkin here.

One thing both NYC and my city (Portland) do right is LPIs: Leading Pedestrian Intervals. At more and more crossings, they've been making it so that the pedestrians get the little walking man about 7 seconds before the cars get the green. Ensuring that drivers turning have to sit there and see that there are pedestrians for them to wait for.

Sometimes they make clicky sounds when you press them

Just like the "close" buttons on elevators. They make you feel better. Often, the buttons don't actually exist, and the "walk" light is on the same system that allows cars to go the same way.