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/codex-mcp/src/connection_manager.rs

@@ -64,7 +64,10 @@ use rmcp::model::Resource;
 use rmcp::model::ResourceTemplate;
 use tokio::task::JoinSet;
 use tokio_util::sync::CancellationToken;
+use tracing::Instrument;
 use tracing::instrument;
+use tracing::trace;
+use tracing::trace_span;
 use tracing::warn;
 
 /// A thin wrapper around a set of running [`RmcpClient`] instances.
@@ -378,13 +381,37 @@ impl McpConnectionManager {
     }
 
     /// Returns all tools with model-visible names normalized.
-    #[instrument(level = "trace", skip_all)]
+    #[instrument(level = "trace", skip_all, fields(mcp_server_count = self.clients.len()))]
     pub async fn list_all_tools(&self) -> Vec<ToolInfo> {
         let mut tools = Vec::new();
-        for managed_client in self.clients.values() {
-            let Some(server_tools) = managed_client.listed_tools().await else {
+        for (server_name, managed_client) in &self.clients {
+            let has_startup_snapshot = managed_client.startup_snapshot.is_some();
+            let startup_complete = managed_client
+                .startup_complete
+                .load(std::sync::atomic::Ordering::Acquire);
+            trace!(
+                server_name = %server_name,
+                has_startup_snapshot,
+                startup_complete,
+                "waiting for MCP server tools while building tool list"
+            );
+            let Some(server_tools) = managed_client
+                .listed_tools()
+                .instrument(trace_span!(
+                    "list_tools_for_server",
+                    server_name = %server_name,
+                    has_startup_snapshot,
+                    startup_complete
+                ))
+                .await
+            else {
                 continue;
             };
+            trace!(
+                server_name = %server_name,
+                tool_count = server_tools.len(),
+                "listed MCP server tools while building tool list"
+            );
             tools.extend(
                 server_tools
                     .into_iter()