I mean, how is anyone going to crytographically verify a video? You either have an icon in the video itself or displayed near it by the site, meaning nothing, fakers just copy that in theirs. Alternatively you have to sign or make file hashes for each permutation of the video file sent out. At that point how are normal people actually going to verify? At best they're trusting the video player of whatever site they're on to be truthful when it says that it's verified.
Saying they want to do this is one thing, but as far as I'm aware, we don't have a solution that accounts for the rampant re-use of presidential videos in news and secondary reporting either.
I have a terrible feeling that this would just be wasted effort beyond basic signing of the video file uploaded on the official government website, which really doesn't solve the problem for anyone who can't or won't verify the hash on their end.
Maybe some sort of visual and audio based hash, like musicbrainz ids for songs that are independant of the file itself but instead on the sound of it. Then the government runs a server kind of like a pgp key server. Then websites could integrate functionality to verify it, but at the end of the day it still works out to a "I swear we're legit guys" stamp for anyone not techinical enough to verify independantly thenselves.
I guess your post just seemed silly when the end result of this for anyone is effectively the equivalent of your "signed by trump" image, unless the public magically gets serious about downloading and verifying everything themselves independently.
Fuck trump, but there are much better ways to shit on king cheeto than pretending the average populace is anything but average based purely on political alignment.
You have to realize that to the average user, any site serving videos seems as trustworthy as youtube. Average internet literacy is absolutely fucking abysmal.