It's less common to have a light or crosswalk away from a junction. Usually when you push the button it changes the lights to the crossing sequence on the next cycle so everything keeps flowing, so a pedestrian might hit the button, then wait for traffic to stop, then cross traffic, then a dedicated turn cycle. It can be a bit of time.
The light in question would turn yellow the moment you pushed the button, without integration with the nearby lights. Driver's usually don't expect to get stopped after being allowed to pass a red light after only about 100 meters.

In general, there's too much variation to say the US as a whole doesn't do some traffic thing. If it exists odd are someone tried using it on a road somewhere.