They would, but each camera's private key can be extracted from the hardware if you're motivated enough.

If Alice's fancy new camera has the private key extracted by Eve without Alice's knowledge, Eve can send Bob pictures that Bob would then believe are from Alice. If Bob finds out that Alice's key was compromised, then he has to guess as to whether any photo he got from Alice was actually from Eve. Having a public timestamp for the picture doesn't help Bob know anything, since Eve might've gone and created the timestamp herself without Alice's knowledge.