Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?
Given the tool's simplicity (no params, no output schema), the description is complete. It explains the return values (authenticated, pending, connect_url) and the conditions for each state. The only minor gap is not explicitly stating that this is a read-only operation, but annotations cover that. For a status-check tool, this is sufficient.
Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.