I didn't get this impression; that is, the impression I got from the video wasn't "you should accept your interlocutor's perspective as the Only True Perspective" but "don't give Weasel Apologies."

Then I believe that you missed it. What you say was in the video (and I mentioned it), but the part about accepting the supposed victim's claims of being hurt as proof that you sinned (your "Only True Perspective" goes a bit too far) is there and is the backbone of the video.

The video's complete list of claims, in a brief form:

1. You will slip up, but when you've done something crappy, you have a choice: either apologize and regain your awesome, or be a fartbag.
2. Analogy: I stepped on your toe, you yell in pain, and I blame you for "standing everywhere".
3. Intent is irrelevant: "I understand you didn't mean to step on their toe, but you still did, and you caused it, so apologize".
4. Don't blame people for how they feel, blame yourself, you've caused it.
5. Do feel bad.
6. When apologizing, don't think you're asking for forgiveness.
7. Figure out what you did wrong, believe it, understand it, internalize it
8. Figure out why you did the hurtful things and provide context.
9. Don't just express sympathy, you have to accept the blame.
10. Tell people you won't do it again.
11. Don't think you're losing or that it's a zero-sum game. Apologizing is a sign of strength.
12. Do in fact change your behavior.

Points 2-4 basically set up the premise that someone's feelings being hurt by your words means you've done something bad and should apologize. You're not allowed to question the appropriateness of those feelings, and you're not allowed to introduce your intent. Your own moral judgement is never mentioned.