You can use a signature to verify that a video hasn’t been modified, but that doesn’t really help in itself with fakes since fakes can be signed too. And if the cameras are doing the signing, you’d just need to extract the private key from the camera and use it to sign fakes. It would require something like the secure enclave to protect the private key and, of course, we’d then be trusting the camera manufacturers.

Not to mention creating a world in which people can't build their own cameras, or change the software on those cameras...

Sure you can. Just wouldn’t be usable as factual proof of an event.

The camera could sign it while recording...

But whats to stop someone reverse engineering the camera and using the signing key to sign something else?

Wouldn't that also allow one to determine the camera with which the video was recorded? Not exactly something a whistle-blower would want.