This is incorrect. The gate agent has nothing to do with it.
I read this post to a UA pilot. He said its all automated and connected to the plane's braking & landing gear systems. When the pilots release the brakes at the gate it sends a signal to Ops. Another signal is sent when the plane lifts off the ground. That's connected to the landing gear movement (not the brakes). That's called the "off time" ... which then calculates the ETA. Same deal with landing (re: landing gear). When the plane is waiting to be waved in, if the pilots set the brakes that sends an 'arrival' signal. After they get to the gate and re-set the brakes, the arrival time is re-set then re-sent when the door opens.

There are newer computers in some planes that won't send the departure signal until the plane actually moves -- ops can tell that by GPS.