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/lib.rs

@@ -15,6 +15,7 @@ pub use rmcp_client::MCP_SANDBOX_STATE_META_CAPABILITY;
 pub use runtime::McpRuntime;
 pub use runtime::McpRuntimeContext;
 pub use runtime::McpRuntimeInput;
+pub use runtime::McpStartupPolicy;
 pub use runtime::SandboxState;
 pub use tool_catalog_cache::McpToolCatalogCache;
 pub use tools::ToolInfo;