Or there is something else?
Your question indicates a lack of awareness between how the exposure time (i.e. "shutter speed") can differ from the shutter transit time (i.e. amount of time it takes to expose the whole picture):
Just because a camera is set to 1/1000 sec does not mean the entire photograph is captured in 1/1000 sec. It means any particular spot on the film/sensor is exposed for 1/1000 sec. But usually the entire frame is exposed sequentially from one side to the other or from top to bottom.
For modern digital cameras with mechanical shutters this typically requires around 2.5-3.5 milliseconds plus the exposure time. (Most current digital cameras with "electronic shutters" take even longer to fully scan the sensor from top to bottom for one cycle, resulting in more severe "rolling shutter" effect.)
Cameras with focal plane shutters have two shutter curtains. The first one opens to expose the sensor/film to light, the second one follows it in the same direction to cover the sensor/film to end the exposure. The shutter curtains move at the same speed regardless of the exposure time used. The exposure time is determined by the interval between the beginning of the movement of the first curtain and the beginning of the movement of the second curtain.
Exposure times shorter than the curtain transit time are accomplished by beginning to close the second curtain before the first curtain has fully transited. The only difference between 1/1000 and 1/2000 is the width of the slit between the two shutter curtains as the second one chases the first across the focal plane.
For a camera with a shutter transit time of 2.5 milliseconds and using a Tv of 1/1000 (1 ms), exposure on one side frame is begun 3.5 milliseconds before exposure on the other side of the frame is ended.
The reason cameras have a flash sync speed is to insure that the first curtain is fully open and the second curtain has not yet begun to close when the flash is fired, and that both curtains remain fully open long enough for the flash to release most of its light energy over a short time interval. Cameras with a flash sync of 1/250 sec typically have a shutter curtain transit time of around 2.5-3 milliseconds and allow the flash another 1-1.5 millisecond to release its energy before the second curtain begins to close. The flash sync signal is sent to the hot shoe a miniscule fraction of a millisecond after the first curtain is fully open. But if you're using an exposure time of 1/1000, then the second curtain is already covering over half the sensor by the time the sync signal is sent.
In your case using a Tv of 1/1000 with a camera that has a 1/250 X-sync, the signal is sent at approximately 2.5-3 milliseconds after one side of the image has begun to be exposed and approximately 1 millisecond before the other side of the image will finish being exposed.