Couldn’t I just change the camera date?

You post it publicly somewhere that has a timestamp. A blockchain would be best because it can't be tampered with.

Still, unique keys for each camera would lessen the risk of someone leaking a single code that undermines the whole system, as happened with DVDs.

And if an interested party wanted to steal a camera's private key to fake an image's provenance they'd need to get physical access to that very camera. Perhaps a state-sponsored group could contrive this (or intervene during manufacturing), but it is a challenge and an even bigger challenge for everyone else.

original image’s timestamp has already been published

"Oh the incorrect information was published, here's the correct info". Again, the map is not the territory.

the whole point of this technology is to remove the need for that trust.

And it utterly fails to achieve that here. I'll put it another way: You have this fancy camera. You get detained by the feds for some reason. While you're detained, they extract your private keys and publish a doctored image, purportedly from your camera. The image is used as evidence to jail you. The digital signature is valid and the public timestamp is verifiable. You later leave jail and sue to get your camera back. You then publish the original image from your camera that proves you shouldn't have been jailed. The digital signature is valid and the public timestamp is verifiable. None of that matters, because you're going to say "trust me, bro". Introducing public signatures via the blockchain has accomplished absolutely nothing.

You're trying to apply blockchain inappropriately. The one thing that publishing like this does is prove that someone knew something at that time. You can't prove that only that person knew something. You can prove that someone had a private key at time X, but you cannot prove that nobody else had it. You can prove that someone had an image with a valid digital signature at time X, but you cannot prove that it is the unaltered original.