Handle required MCP startup failures across components (#10902)

Summary
- add a `required` flag for MCP servers everywhere config/CLI data is
touched so mandatory helpers can be round-tripped
- have `codex exec` and `codex app-server` thread start/resume fail fast
when required MCPs fail to initialize

codex-rs/exec/tests/suite/mcp_required_exit.rs

@@ -0,0 +1,38 @@
+#![cfg(not(target_os = "windows"))]
+#![allow(clippy::expect_used, clippy::unwrap_used)]
+
+use core_test_support::responses;
+use core_test_support::test_codex_exec::test_codex_exec;
+use predicates::str::contains;
+
+#[tokio::test(flavor = "multi_thread", worker_threads = 2)]
+async fn exits_non_zero_when_required_mcp_server_fails_to_initialize() -> anyhow::Result<()> {
+    let test = test_codex_exec();
+
+    let config_toml = r#"
+        [mcp_servers.required_broken]
+        command = "codex-definitely-not-a-real-binary"
+        required = true
+    "#;
+    std::fs::write(test.home_path().join("config.toml"), config_toml)?;
+
+    let server = responses::start_mock_server().await;
+    let body = responses::sse(vec![
+        responses::ev_response_created("resp_1"),
+        responses::ev_assistant_message("msg_1", "hello"),
+        responses::ev_completed("resp_1"),
+    ]);
+    responses::mount_sse_once(&server, body).await;
+
+    test.cmd_with_server(&server)
+        .arg("--skip-git-repo-check")
+        .arg("--experimental-json")
+        .arg("tell me something")
+        .assert()
+        .code(1)
+        .stderr(contains(
+            "required MCP servers failed to initialize: required_broken",
+        ));
+
+    Ok(())
+}