Start cached MCP servers lazily for subagents (#37261)

## Why

Subagents can reuse cached MCP tool definitions without starting every optional
server before any of its tools are used.

## What changed

- Defer startup for subagent-owned, optional MCP servers when their cache has at
  least one visible, allowed tool, and trigger startup on the first operation
  that needs the live client.
- Keep root sessions, required servers, selected plugin servers, and explicit
  reconnects eager.
- Exclude empty catalogs from cache eligibility and keep dormant servers out of
  the initial startup summary.

## Testing

- Extend MCP tool-cache coverage for eager root startup, unused and filtered
  subagent servers, deferred startup status events, and startup that survives an
  interrupted first tool call.

GitOrigin-RevId: e7528ff08312583a6747cbc6550ea5b6eabda9a6

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

@@ -316,10 +316,10 @@ impl ManagedClientStartup {
             .startup_timeout_sec
             .unwrap_or(DEFAULT_STARTUP_TIMEOUT);
         let cancel_token_for_fut = cancel_token;
-        let tool_catalog_fetch_ticket = tool_catalog_cache_context
-            .as_ref()
-            .map(McpToolCatalogCacheContext::begin_fetch);
         async move {
+            let tool_catalog_fetch_ticket = tool_catalog_cache_context
+                .as_ref()
+                .map(McpToolCatalogCacheContext::begin_fetch);
             let refresh_start = is_codex_apps_mcp_server.then(Instant::now);
             let outcome = match async {
                 if let Err(error) = validate_mcp_server_name(&server_name) {