Adding signature after recording works to certain degree, but it still does not guarantee that the content is what the camera saw.
Cameras can record screens.
Perhaps adding a signed channel for depth and/or non-visible light would be the next step.
Or you could just smear some Vaseline on the lens and tell people the lens got dirty. It hurts the credibility for anyone who knows about these cameras but I doubt the public would think about it that much.
You're pushing a (bad) technical solution to a social problem.
Cameras that cryptographically sign their output will not solve anything. The idea has more flaws than it's possible to list, but here's a big one: do you really think a technological gimmick like that would stand up to a nation state? Do you really think the CIA, NSA, FSB, Chinese Ministry of State Security, etc. will not be able to sign whatever the hell image they want with a camera's signature?
Also, how to protect a chip from reverse engineering even from all except "top actors"? I remember the price for reverse engineering of certain ICs was between 5 and 7 figures of USD. Don't know about modern IC processes, but it may be affordable for many even for those?
But even hardware can be hacked/bypassed if the effort is worth it.
I don't know audio well enough how it happens there. But potentially it can be signed in chunks as well.
Of course, one needs to consider risks if editing can make content appear different than originally intended, when the video as "whole" is not signed. But for that, different entity can be used again.
It would require completely changing how software currently handles video editing in short.
At the beginning, camera manufacturers might need to provide their own editors, to make editing possible. How much we can trust the camera holders, if the editor software even allows using the key from the camera for better editing in certain limits?
If you just cut away to the original clip and didn't have any modifications like motion graphics over the top of it you could in theory pass through the original video with the same compression and signing without too much drama but any modifications over that or presenting it as picture in picture would be a big difference as now you need to have both the original frames with the added graphics on top.
Does that really matter? You could sign keyframes, and then also sign the differences frame by frame til you get to the next keyframe.
I wonder if you could use the camera to record deepfaked video and in effect bless a lie. Even just filming a TV set might be enough for low grade blackmail and much more complicated methods are available.
For example the whitehouse is known to revise the text of the president’s speeches when he says the wrong thing. If we only have officially released videos where the gaffes and fables are left out, how is anyone to know what he actually said?
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.
For example, when watching a video on YouTube containing a speech by the president (provided on an official channel like the White House's), there should be a clear indication that the video has a digital signature and the option to verify it on an independent government website.