The elbows will go where they need to. If that isn't below the hands, it's because the hands are in the wrong place.
That said, it couldn't hurt to pack your scapula in tighter (which allows your lats to stabilize your humeri more effectively). Watch what happens to your shoulder between seconds 9-11 in the video: you have your scapula adducted correctly at second 9, and then kind of shrug your shoulders up to get them closer to the bar (almost certainly as an unconscious part of unpacking).
This is handy because it means that if you are correctly positioned, unracking the bar should feel noticeably less comfortable than it does now. Ensure your shoulders do not move from the point you set them with your "leg drive." Getting your spotter to help you with your liftoff may also help