For the 1,000th time, it’s perfectly fine for a public speaker or presenter to put one hand in pocket.
No, it’s not “unprofessional.”
No, it doesn’t mysteriously “direct” audience eyes downward.
The “no hand in pocket” is part of that oral tradition of bad and vague presentation advice that seems to have taken on a life of its own.
Passed from person to person in anecdotal fashion.
Who knows where or why these things originate? Certainly not from the 2,500 years of public speaking literature.
While it may not contribute to your personal competitive advantage, it can bestow a message of calm competence.
And who doesn’t want that?