Is it okay to replace "shut" with "closed"? I think "closed" is more appropriate considering that he is speaking in a formal speech situation. Doesn't "shut" sound too rude?
You're right, it would sound a little less rude to say "kept your mouth closed" instead of "kept your mouth shut" here. Both "closed" and "shut" sound pretty informal but "shut" is a little more rude. However in this case, he is not speaking very formally throughout the presentation and his tone of voice is very casual, so it seems like he speaks this way on purpose in the video.