I've been told this as well (recently, actually) but not from any of the teachers I had at hombu.
The analogy I was given alot at hombu was that when you try to push against a wall or push a car up a hill or something like that you need to tilt your hips down to take pressure off your lower back and to get the most power from your back foot in order to push. I actually just tried this against the wall here in my office...luckily no one was looking
Also...there's a difference between tilting the hips down so that your butt sticks out and tilting your hips down so that everything becomes a straight line. This is what Takeno Sensei's manipulation of my kamae showed me. Perhaps tilting the hips down is the wrong way to say it...perhaps a better way would be to say "roll your hips forward until they lock in line with your back leg and the hips become in-line with (part of? welded to?) the plane that your back leg makes with the joint at the hips.
Unfortunately, I haven't trained in any of the grappling arts to see the contrast. My suspicion is that the premise that I have been told that Aikido takes is that you apply all your power at a single point at a single time and that this hip tilt helps that happen. It is possible...and again I don't know...that other systems emphasize something else (perhaps ease in moving in any direction?) which the hip tilt would not help with.
WRT point (2)...although Kamae is a training stance, I don't think it would emphasize something so strongly if it weren't one of the key things we were supposed to take away from it. Same with the "weight forward" idea and the circle that your arms make to put them in front of you being where they are strongest.