If you are talking about the break at 1:40 into the video, that is actually a cloud layer. Not a break. You can see the shuttle fly through it at :42.
The other shuttle videos were filmed at 3 miles or less from the launch point. The audio from the launch is intense. But the spread of the exhaust at ground level is less than a mile wide and barely high enough to cover the launch tower. I'm not sure how far out people who believe a missile was launched off the coast of California believe it was launched, but the spread of the trail starts above the haze layer you've been talking about, and extends for miles to the south of the trail. Nothing at all like the exhaust from the solid rocket boosters and main engines of the largest spacecraft we've launched in decades (if not ever).
And I found the sound barrier break. Very cool.
I wasn’t putting them out there as exact comparisons. No one has done that with either a missile contrail or an airplane contrail. Just as examples of how fast a missile contrail can expand and distort. At about 1:40 - 1:41 there is a big break in the contrail right above the wing of the plane. It only lasts a second because he moves the camera.