How would the server know that the request is coming from a real iPhone?

This is a pretty standard application of trusted computing and can be done entirely on the iPhone. A server would only possibly be needed for anonymization (while retaining key revocation capabilities if a key does end up leaking), but there are serverless ways to do even that (TPMs have supported these for a while now).

I think there's a chain of trust. The sensor signs raws, and the private cloud takes signed raws, does minimal processing so they're at least coherent, and re signs that output (maybe even including the original signed raw as well in the image file).

That's a good point, You might still be able to trick the cloud to sign your photos, but that's something they could patch in updates without losing control of the key. They could have the server only sign photos taken on the latest ios version.