You only sign what you want to sign.

When you publish a video of you speaking at a public event, you sign it. When someone else publishes a doctored video of you, you do not sign it.

This alone doesn't protect you in the case that someone speaks and then later intentionally doctors and signs the video in order to change what they said. In this case trusted third parties (e.g. news organizations) could sign videos as well. A set of signatures taken together can provide trust.

That's not how this works though. If I want to post video clamining someone did something, I sign it, but what does this prove? Nothing.

Well yeah, because you aren't the subject of the video and you have no reputation.

You wouldn't think a letter from your mom is from your mom unless your mom signed it.

It's not supposed to validate the truthfulness of the content, why would you think that was the purpose?