Most videos people watch nowadays don't come from "official" authorities, whose public keys are known and signatures can be verified. When videos are recorded by random people from their smartphones, what signature are you going to verify?

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.

I really don't see how crypto can solve this problem. I don't think AI can either (for reasons already mentioned in this thread). It's something we'll have to learn to live with.