Renzuki (two or three rapid punches in a sequence) is a real technique that is quite effective. It is Shotokan's version of the Wing Chun "chain punch." Ours is slower than theirs, but a lot more powerful.
The problem is that renzuki is dramatic, and therefore someone used it to liven up various katas. The glaring example is in the middle of Empi kata. The move where you find yourself staring into the palm of your open left hand is the beginning of a textbook te kagami wristlock and throw, exactly as taught in hakkoryu jujutsu. Unfortunately, just as you are about to execute the throw, the kata has you drop the miscreant's hand and do renzuki, then grab his hand again and finish the throw.
It's pretty clear that somebody stuck the renzuki in to liven up a "boring" part of the kata.
The lesson is to be suspicious of renzuki in katas. It is too easy to spice up a kata by putting in some fast punches. If you have an interpretation that works except for the renzuki in the middle, just set the punches aside. They are a shotoism.