Sure, but aljazeera can sign a deepfake with their key just as readily as a real video, the signature does not ensure the realness of the video - it can only establish a part of it's "chain of sources".
But are you going to trust aljazeera's signature on a [potential] deepfake of an American politician as the only source of truth about whether that video is real?
In a technical sense, signing only attests that a specific key pair was used to process the data. How that key pair is associated to its handler is a social function. If the signer claims authenticity, the general population will accept the claim, regardless of whether the claim has technical merit.
We can do a full "Video SecureBoot" thing, where the camera sensor signs stuff at the lowest level and every other level adds additional signatures, so you know the full chain of what happened (what cam took what part of the final video, what software edited, and who did the editing).
Nothing in that scheme requires "rights management".
Ultimately still it comes down to trust: do you trust the org who put out the video, do those people trust their subcontractors and (freelance) videographers, the hardware and software that was used, etc
In some cases this might help a little. But in the end misinformation and outright lies will still spread, be it deepfakes or be it just the written word. QAnon is relatively successful without the need to deepfake (they will just do a few meme pictures), so is the Flat Earth movement, and Alex Jones (US not UK) has still a following too, for example.
Several years ago I researched and worked on this concept in my spare time in the hope of creating an alternative to DRM and an alternative revenue model.
Instead of forcing users to use authorized devices and checking if they are allowed to consume the content I wanted to encourage users to desire authentic digital content. This would have involved a container capable of holding chains of signatures, PKI, and players with a feature similar to the padlock in browsers.
Authentic for pay content would have contained unique signatures that acted as a watermark.
If someone ilegally shared the signed version it could be traced back and action could be targeted. The goal was not to make it impossible to pirate content (I find it foolish to try to prevent this) but to make unsigned content undesirable due to uncertain authenticity. The same way authentic colectables are desirable due to their authenticity. A forgery of a famous painting is worth a lot less than the original if you know it is fake even if it is a perfect copy. A reissue of a first edition comic book is worth less even if the content is the same.
In some markets 3C marketing works better than 4P marketing and this results in a circular symbiotic relationship between creator and consumer rather than a hierarchical antagonic relationship.
This also included a way of remixing content that allowed distribution of the original along with the alterations. A sort of open source for media. Think of distributing your Blender/Premiere/ After Effects/etc. project file instead of the rendered result. It would have also been possible to share the alterations without the original and let the user provide their own copy if the original.
This was based on a very wrong understanding of what is a derivative from a legal pov.
Alas, this was a foolish dream in my youth, but it was a vision of what media formats would have been like in a parallel world where authenticity was worth more than monopoly over distribution. A world where plagiarism would have been a bigger crime instead of copyright infringement.
