You have to be at the gate 20 minutes before departure time if you want to be on board. In general for UX flights, they will wait until 10 minutes before departure before they close the door in an effort to get the flight to push back on time. If you saw the SF3 props spinning, you were too late.

As for compensation, since you were late into IAD because of UA, I would expect some token gesture. They still got you there that night. If you wanted to get to HPN and not LGA, then you could have got the first flight out in the morning and UA would have comped you meals & a hotel. I'd be surprised if you got more than a $100 CS cert, but stranger things have happened.

Myself, I wouldn't bother asking for compensation and would just remember not to book such a tight connection onto the last flight of the night. If you get anything from this experience, let's hope it was an education.