Videos by third parties, like Trump's pussy grabber clip, would obviously have to be signed by them. After having thought about it, I believe this is a non-starter.

It just won't be as good as https. Such a signing scheme only makes sense if the media is shared away from the original website. That means you can't just take a quick look at the address bar to make sure you are not getting phished. That doesn't work if it could be any news agency. You have to make sure that the signer is really a trusted agency and not some scammy lookalike. That takes too much care for casual use, which defeats the purpose.

Also, news agencies don't have much of an incentive to allow sharing their media. Any cryptographic signature would only make sense for them if directs users to their site, where they can make money. Maybe the potential for more clicks - basically a kind of clickable watermark on media - could make this take off.