Then, your agent invents a User-Agent and it sometimes invents one with your email in it.
It doesn't even require prompt injection or a failed request, it might be in the agent's pretraining knowledge that a given website requires a user-agent.
It's just doing what it's supposed to do, analyzing the statistical patterns of a natural language prompt and matching it to a likely response.
What's ridiculous is an assumption that it "it should never do something so ridiculous" as if an LLM is a person who understands what that word means in any context other than next-token prediction.
Where Claude Code has supposed boundaries between deterministic and non-deterministic tasks and flags.. generally.. it's unclear how much privacy has been a part of that and what implication that could have.
Hope that clarifies.