Yes, it would make sense in most cases to grab your alternate. My theory is that you were not taught to do so because when the skill was first taught, there were no alternates, and they still haven't changed the method.
As for the sweep method being the easiest and most reliable, I disagree, and why I disagree is a long story.
Decades ago almost all instruction on basic skills was done with students kneeling on the bottom of the pool or in the sand. If you learn your skills that way, the way you do those skills is completely different from the way you do them on a real dive, with your body in a horizontal position. You are learning one way to do things in class, but in the real world it is different. In the last few years, instructors around the world have been switching to teaching skills with students in a horizontal diving posture, and it makes a huge difference.
Sweep: If you are kneeling with the body vertical, leaning to the right is different from leaning to the right while horizontal. The regulator will hang differently as well. It is actually easier to do it while horizontal because the regular will naturally fall in front of you.
Reach: When you are kneeling and vertical, gravity takes the tank valve and the regulators down and away from you. It is so hard to reach the hose that you are usually taught to lift the bottom of the tank with your left hand to get it close enough to reach it. It is still hard to reach it. That is why most people say the sweep method is so much easier. On the other hand, if you are horizontal, gravity puts the tank valve and regulator right behind your head. Your regulator hose is right behind your ear. It is extremely easy to reach the hose in that position, much easier than the sweep method.