Governments are very trigger happy on regulation, like I said this is an impossible problem to solve without public/private key verif (or alternative).
Embedding a key might help technologically distinguish a small subset of the videos we're worried about but 1) ok so then what, you still need the regulations that make impersonation not allowed which is what these are 2) what about unofficial sources, hot mikes and leaked tapes etc.
Basically, you're suggesting a technological approach that's fine for a narrow set of concerns, but you still have to regulate and disallow behaviors that can be prosecuted. The actual problem here is not a solved problem, but yes in one very narrow subset of problems, a more complicated solution than hosting official videos on official sources would be to require official sources to also sign the video.
An easier requirement from official sources to embed a public key in a video. This is a solved problem