Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?
Given the tool has no parameters, no output schema, and annotations cover safety, the description is fairly complete. It explains the two possible return states and what they mean. It could be slightly more explicit about the exact structure of the response (e.g., fields like authenticated, pending, connect_url) but overall it is adequate for a status-checking tool.
Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.