If gesturing is a more effective way of speaking, then it make sense for the power structure to take it away from the subordinates. If a private could convince his superior to believe something or do something then it upsets the hierarchy. Maybe there's unscientific words like hypnosis or mesmerization you could attach to this idea of gesturing with speaking.

In D&D, there's somatic, verbal, and material components to spell casting. Somatic components are the precise hand movements to aid in spell casting. That fictional idea/meme had to have been generated by some real-world phenomena, no?

Edit to expand: I also think this may be why even beyond the military there seems to be a widespread stigma against gesturing while speaking. Not that a majority of people look down on it, but that some significant percentage tend to do so at least.

It might seem unfair to some parties to have these extra gestures "used" on them to convince them of things that go beyond what the words they use actually mean. I know personally, I much much prefer to read what politicians said, via a transcript, rather than watch them speak. Their body language + gestures + tones can really change perceptions so significantly that I don't want to let myself be preventably persuaded by otherwise disagreeable positions.