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.
The answer is that it doesn't. Some might try to make it work by using proprietary video editing software that signs a ledger of what edit operations were performed, or something like that, but that doesn't work. The signing keys will eventually be extracted from the video editor or the camera, or the video editor or camera will be hacked to sign something it shouldn't. You might say that this at least stops low-skilled attackers, but the misinformation that is most dangerous to humanity, that created by governments to start wars, won't be impeded by any of these schemes. The whole cryptographic signature proposal is worse than useless.