If you wish, I could go on to explicitly analyze this video bit by bit in order to achieve a more calm presentation of the ideas therein. It would go something like: "Frequent mistake when apologizing, then how to do it right"

Yes, that kind of format would be a much better way to present the ideas. It would then be easier to have a constructive argument about them. Provided, of course, that the reason is given for why said mistakes are actually mistakes.

But anger and "browbeating" are perfectly legitimate ways to present something that you care very much about; most of the Sequences are written in such a lecturing, passionate tone.

The tone of the sequences is far from emotionally neutral, and 'passionate' would be a good word to describe many parts of the sequences, but the way is this is often done is by providing a justification, building up to the main point, then using a passionate form of presentation to emphasize the important take-away points. It seems that this video skips right to the 'passion' part without addressing the 'why we should even care' part.

And that seems like the most important part to me. If you don't mind me saying, I get the impression that you might be missing that part as well. Politeness is all about human psychology and interaction and our in-built sense of empathy towards others. The correctness of an argument has little to do with how politely it is presented, but its impact on other people, on the other hand, does. So it would be useful examining this a little bit more deeply. Again, this is something that is often done in the sequences: things are broken down deeply and a 'view from above' is taken.