[codex] trace MCP startup latency (#28630)

## Summary

- add trace-level instrumentation around per-server MCP setup, client
construction, initialization, and initial tool listing
- trace Codex Apps tool and server-info cache loads
- attach `server_name` to server-scoped spans so slow startup work can
be attributed to a specific MCP server

## Why

`session_init.mcp_manager_init` can occasionally be slow, but its
existing coarse span does not identify whether time is spent loading the
Codex Apps cache, constructing a client, initializing a transport, or
listing tools. These definition-level spans provide that breakdown
without changing startup behavior.

## Validation

- `just test -p codex-mcp` (87 passed)
- `just test -p codex-rmcp-client` (86 passed, 2 skipped)

codex-rs/codex-mcp/src/rmcp_client.rs

@@ -67,6 +67,8 @@ use rmcp::model::JsonObject;
 use rmcp::model::ProtocolVersion;
 use rmcp::model::Tool as RmcpTool;
 use tokio_util::sync::CancellationToken;
+use tracing::Instrument;
+use tracing::instrument;
 use tracing::warn;
 
 /// MCP server capability indicating that Codex should include [`SandboxState`]
@@ -141,6 +143,7 @@ pub(crate) struct AsyncManagedClient {
 impl AsyncManagedClient {
     // Keep this constructor flat so the startup inputs remain readable at the
     // single call site instead of introducing a one-off params wrapper.
+    #[instrument(level = "trace", skip_all, fields(server_name = %server_name))]
     #[allow(clippy::too_many_arguments)]
     pub(crate) fn new(
         server_name: String,
@@ -228,7 +231,7 @@ impl AsyncManagedClient {
             startup_complete_for_fut.store(true, Ordering::Release);
             outcome
         };
-        let client = fut.boxed().shared();
+        let client = fut.in_current_span().boxed().shared();
         if cached_tool_info_snapshot.is_some() {
             let startup_task = client.clone();
             tokio::spawn(async move {
@@ -306,6 +309,7 @@ impl From<anyhow::Error> for StartupOutcomeError {
     }
 }
 
+#[instrument(level = "trace", skip_all, fields(server_name = %server_name))]
 pub(crate) async fn list_tools_for_client_uncached(
     server_name: &str,
     is_codex_apps_mcp_server: bool,
@@ -529,6 +533,7 @@ fn validate_mcp_server_name(server_name: &str) -> Result<()> {
     Ok(())
 }
 
+#[instrument(level = "trace", skip_all, fields(server_name = %server_name))]
 async fn start_server_task(
     server_name: String,
     client: Arc<RmcpClient>,
@@ -658,6 +663,7 @@ struct StartServerTaskParams {
     supports_openai_form_elicitation: bool,
 }
 
+#[instrument(level = "trace", skip_all, fields(server_name = %server_name))]
 async fn make_rmcp_client(
     server_name: &str,
     server: EffectiveMcpServer,