We don't have to live in the world where people are maximally naive (even if it seems so today). That also assumes there's not a signed video available of the event, usually things are recorded by more than one person especially a speech by the president.

The biggest risk IMO is that key becomes immediately one of the most important secrets to keep since it holds the promise of validating anything you want to lie about.

Signing keys can be derived from a root key that allows for rotation and revocation. Multiple keys can sign the same content.

Now you have to securely deliver those keys to the cameras and people have to keep them up to date. With smartphones it's a bit easier because that can just be pushed to the phone automatically but for news orgs and other professional outfits their camera's aren't internet connected. So then you have a weird mishmash of deciding if an out of date key is being used because it's been cracked/stolen or if the NBC stringer just didn't update their camera before heading to the event.