I work in New York, but I go to Seattle periodically because that's where my company's headquarters is. How people treat the buttons is wildly different between those two cities. I hardly ever see anyone press the cross button in NY, since the convention is to wait for there to be no cars and then cross (and not bother with the button). In Seattle, I've seen people wait patiently at a cross walk even though there's no cars anywhere in sight.

I would imagine rather than a placebo effect, it's more of a self-selection phenomenon - rather than pressing a button making people more patient, the people who are more likely to follow "the rules" are more likely to press the button. And even if there is hypothetically a placebo effect, the result would be irrelevant if almost no one presses the button.