> Most videos people watch nowadays don't come from "official" authorities

I doubt this is true. Most videos are likely disseminated from a handful of companies.

> Even if smartphone manufacturers start integrating digital signatures right into their cameras, you can use the smartphone to re-record a pre-recorded fake video. And I'm sure that with enough resources you can do something way more clever.

Even if a fake video was re recorded it would have a different signature than the “original”. Problem solved.

Literally cryptography is the only solution