A crash involving whom? I'd be shocked if a driver can hit a jaywalking pedestrian and not be at fault.
The idea is that the higher purpose of a $1 ticket is creating a paper trail of jaywalking behavior that could be useful in court.
It can definitely happen. I'm guessing it varies state by state, but I was hit by a car while out running. I was crossing at a green light, didn't press the button, and a car made a left turn at the same time and hit me. The police report said the driver was not at fault.
Massachusetts pedestrians are only guaranteed right of way in marked crosswalks that do not have a signal. When there is a signal the light denotes right of way.