Actually, the technique you describe against a boxing type punch (very similar to a Yokomenuchi in Aikido, or what I refer to as a "John Wayne" punch) is almost the same as a Sumiotoshi or "Corner drop throw" in Aikido. For the sake of argument here, not all Aikido dojo refer to this type of defense as Sumiotoshi; so I'm asking everyone on the forum not to respond by saying that I've described the wrong technique as I realize that your particular dojo/organization may use different terminology. The nage would enter into the direction of the punch just as you describe, but instead of using the other hand to deliver a strike, the nage would use his/her other hand to apply downward pressure on uke's elbow, taking him/her off balance and plopping them on the ground. Of course, timing is again everything with respect to this technique. It's just an idea, take it for what it's worth.