Refactor Logging Impl

Loop on stop_token and remove final_entry in Entry.
Move Backend thread out of Impl Constructor to its own function.
Add Start function for backend thread.

src/common/logging/backend.cpp

@@ -223,17 +223,15 @@ private:
     }
 
     void StartBackendThread() {
-        backend_thread = std::thread([this] {
+        backend_thread{std::thread([this] {
             Common::SetCurrentThreadName("yuzu:Log");
             Entry entry;
             const auto write_logs = [this, &entry]() {
                 ForEachBackend([&entry](Backend& backend) { backend.Write(entry); });
             };
             while (!stop.stop_requested()) {
-                entry = message_queue.PopWait(stop.get_token());
-                if (entry.filename != nullptr) {
-                    write_logs();
-                }
+                entry = message_queue.PopWait();
+                write_logs();
             }
             // Drain the logging queue. Only writes out up to MAX_LOGS_TO_WRITE to prevent a
             // case where a system is repeatedly spamming logs even on close.
@@ -241,7 +239,7 @@ private:
             while (max_logs_to_write-- && message_queue.Pop(entry)) {
                 write_logs();
             }
-        });
+        })};
     }
 
     void StopBackendThread() {