code-mode: Remove Session::is_alive() (#29732)

Remove this unused API. This API is insidious in that it implies that
alive state should be determinable from the caller, and implies that a
preflight should indicate routing. Lets drop this, and handle errors
correctly from a failed session in the future.

codex-rs/code-mode/src/session_runtime/mod.rs

@@ -63,10 +63,6 @@ impl<D: SessionRuntimeDelegate> SessionRuntime<D> {
         }
     }
 
-    pub(crate) fn is_alive(&self) -> bool {
-        !self.inner.shutdown_token.is_cancelled()
-    }
-
     pub(crate) async fn execute(
         &self,
         request: CreateCellRequest,

codex-rs/code-mode/src/session_runtime/tests.rs

@@ -152,7 +152,6 @@ async fn shutdown_rejects_cell_admission_queued_before_the_registry_lock() {
     })
     .await;
 
-    assert!(!runtime.is_alive());
     drop(cells);
     assert!(matches!(execution.await, Err(Error::ShuttingDown)));
     assert_eq!(shutdown.await, Ok(()));