Hank, the weight in the Goblet Squat is more than just weight on the stretched position - it's a counter-balance to the tendency to fall backwards and the tendency to round the torso in response to a fear of falling backwards.
You can apply the principle - of using a weight to help with a stretch - to some powerlifts or their variations, e.g., dumbbell bench press, barbell back squats, but these things are only for those whose technique is solid and have very clear understanding of what is and isn't an appropriate weight to use. You would, for instance, never want to do this in a barbell back squat if you have a tendency to round your lower back at the bottom.
Another example, and I believe this comes from the original Russian Kettlebell Challenge book, is to pause at the top of your kettlebell military press and lean forward from the shoulder. It's a sort of standing arm bar. Note that you don't let the weight move backwards, you just press forward from the shoulder.
-S-