simplify: use static local singleton pattern for root session ID

Match the codebase's existing singleton pattern (static local variable).
No mutex or call_once needed.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

Author: khanayan123

src/datadog/root_session_id.cpp

@@ -1,35 +1,21 @@
 #include "root_session_id.h"
 
-#include <mutex>
-
 namespace datadog {
 namespace tracing {
 namespace root_session_id {
 
 namespace {
 
-std::mutex& mutex() {
-  static std::mutex m;
-  return m;
-}
-
 std::string& instance() {
   static std::string id;
   return id;
 }
 
-bool& initialized() {
-  static bool init = false;
-  return init;
-}
-
 }  // namespace
 
 void set(const std::string& id) {
-  std::lock_guard<std::mutex> lock(mutex());
-  if (!initialized()) {
+  if (instance().empty()) {
     instance() = id;
-    initialized() = true;
   }
 }