In all seriousness, this is mostly just one of those defense in depth kind of things. Data exfiltration via a compromised provider is also a very different class of vulnerabilities than prompt injection via malicious tool calls.
So I don't really see this as useful, except in the very limited use case of using a third party proxy who is trying to cheat you (in which case why are you sending all your data to someone you can't trust?).
I think that's where your assumption differs from stated claims.
According to Anthropics own docs[0], they don't alter the models at all after release. So for each model release you should be able to prove that a response was created with that exact model without dumbing it down.
Zero knowledge proofs also still allow you to have disclosed inputs. You can make one of the inputs be a publicly commited hash of the model weights, and have the ZKP validate that the inference was done with that committed set of model weights, in addition to all the inferences in the batch all coming from the same model.
[0]: https://platform.claude.com/docs/en/about-claude/models/mode...