Cryptography is nice here, but the base idea remains the same: you need to trust the person publishing the video to believe the video. Cryptography doesn't help for most interesting cases here, though it can help with another level, that of impersonation.
Sure, Tim Cook can sign a video so I know he is the one who published it - though watching it on https://apple.com does more or less the same thing. But if the video is showing some rockets hitting an air base, the cryptography doesn't do anything to tell you if these were real rockets or its an AI-generated video. It's your trust in Tim Cook (or lack thereof) that determines if you believe the video or not.
All this talk of trust speaks to the larger issue here too - that we've lost so much trust in governments and other important institutions. I'm not saying it was undeserved, but it's still an issue we need to fix.
Sure, Tim Cook can sign a video so I know he is the one who published it - though watching it on https://apple.com does more or less the same thing. But if the video is showing some rockets hitting an air base, the cryptography doesn't do anything to tell you if these were real rockets or its an AI-generated video. It's your trust in Tim Cook (or lack thereof) that determines if you believe the video or not.