Reduce burst testing flake (#9549)

## Summary

- make paste-burst tests deterministic by injecting explicit timestamps
instead of relying on wall clock timing
- add time-aware helpers for input/submission paths so tests can drive
the burst heuristic precisely
- update burst-related tests to flush using computed timeouts while
preserving behavior assertions
- increase timeout slack in
shell_tools_start_before_response_completed_when_stream_delayed to
reduce flakiness

codex-rs/core/tests/suite/tool_parallelism.rs

@@ -69,9 +69,9 @@ async fn build_codex_with_test_tool(server: &wiremock::MockServer) -> anyhow::Re
 }
 
 fn assert_parallel_duration(actual: Duration) {
-    // Allow headroom for runtime overhead while still differentiating from serial execution.
+    // Allow headroom for slow CI scheduling; barrier synchronization already enforces overlap.
     assert!(
-        actual < Duration::from_millis(750),
+        actual < Duration::from_millis(1_200),
         "expected parallel execution to finish quickly, got {actual:?}"
     );
 }
@@ -308,7 +308,7 @@ async fn shell_tools_start_before_response_completed_when_stream_delayed() -> an
     let args = json!({
         "command": command,
         "login": false,
-        "timeout_ms": 1_000,
+        "timeout_ms": 5_000,
     });
 
     let first_chunk = sse(vec![
@@ -371,7 +371,7 @@ async fn shell_tools_start_before_response_completed_when_stream_delayed() -> an
     let _ = first_gate_tx.send(());
     let _ = follow_up_gate_tx.send(());
 
-    let timestamps = tokio::time::timeout(Duration::from_secs(1), async {
+    let timestamps = tokio::time::timeout(Duration::from_secs(5), async {
         loop {
             let contents = fs::read_to_string(output_path)?;
             let timestamps = contents