fix(core): instrument stalled tool-listing handoff (#24667)

## Why

When a turn needs a follow-up request after tool output is recorded,
Codex can still appear stuck in `Thinking` before the next `/responses`
request is opened. The existing local trace showed the last completed
response and the absence of a new backend request, but it did not show
whether the stall was in tool-router preparation or later request setup.

Issue: N/A (internal incident investigation)

## What Changed

Added trace spans around the pre-stream tool-router handoff in
`core/src/session/turn.rs`, including the `built_tools` phase and the
MCP manager read lock.

Added per-server MCP tool-listing spans and trace breadcrumbs in
`codex-mcp/src/connection_manager.rs` with startup snapshot /
startup-complete state so a pending MCP client is visible in feedback
logs instead of looking like a silent hang.

## Verification

- `just fmt`
- `just test -p codex-mcp`
- `just test -p codex-core` (prior full rerun fails in this workspace on
unrelated integration tests: code-mode output length expectations, one
shell timeout formatting assertion, and shell snapshot timeouts; latest
review-fix rerun compiled and passed 1160 tests before I stopped the
abnormally slow unrelated suite)

codex-rs/core/src/session/turn.rs

@@ -1006,12 +1006,25 @@ async fn run_sampling_request(
     clippy::await_holding_invalid_type,
     reason = "tool router construction reads through the session-owned manager guard"
 )]
+#[instrument(level = "trace",
+    skip_all,
+    fields(
+        turn_id = %turn_context.sub_id,
+        model = %turn_context.model_info.slug,
+        apps_enabled = turn_context.apps_enabled()
+    )
+)]
 pub(crate) async fn built_tools(
     sess: &Session,
     turn_context: &TurnContext,
     cancellation_token: &CancellationToken,
 ) -> CodexResult<Arc<ToolRouter>> {
-    let mcp_connection_manager = sess.services.mcp_connection_manager.read().await;
+    let mcp_connection_manager = sess
+        .services
+        .mcp_connection_manager
+        .read()
+        .instrument(trace_span!("read_mcp_connection_manager"))
+        .await;
     let has_mcp_servers = mcp_connection_manager.has_servers();
     let all_mcp_tools = mcp_connection_manager
         .list_all_tools()