## Why
The timeout fallback test depends on short wall-clock deadlines and a server-side sleep, making its timing dependent on thread scheduling.
## What changed
Use a paused Tokio clock and advance past the timeout only after the server receives `HEAD`. Keep the blocking server task alive until the probe consumes the `GET` response to prevent automatic clock advancement during socket I/O.
Assert both request lines and retain the check that the fallback returns `HTTP 405`.
GitOrigin-RevId: da7b2fa9c67e662648da8568ce7f8cfde153dc12