Qt: Simplify Fullscreen UI state tracking

Fixes application closing if the system was shut down while fullscreen.

src/core/fullscreen_ui.cpp

@@ -602,8 +602,6 @@ bool FullscreenUI::Initialize()
   s_state.initialized = true;
   s_state.hotkey_list_cache = InputManager::GetHotkeyList();
 
-  Host::RunOnCPUThread([]() { Host::OnFullscreenUIStartedOrStopped(true); });
-
   if (s_state.current_main_window == MainWindowType::None && !GPUThread::HasGPUBackend() &&
       !GPUThread::IsGPUBackendRequested())
   {
@@ -830,8 +828,6 @@ void FullscreenUI::Shutdown(bool clear_state)
 
   DestroyResources();
   ImGuiFullscreen::Shutdown(clear_state);
-  if (s_state.initialized)
-    Host::RunOnCPUThread([]() { Host::OnFullscreenUIStartedOrStopped(false); });
 
   s_state.initialized = false;
   s_state.tried_to_initialize = false;

src/core/fullscreen_ui.h

@@ -82,9 +82,6 @@ namespace Host {
 
 #ifndef __ANDROID__
 
-/// Called whenever fullscreen UI starts/stops.
-void OnFullscreenUIStartedOrStopped(bool started);
-
 /// Requests shut down and exit of the hosting application. This may not actually exit,
 /// if the user cancels the shutdown confirmation.
 void RequestExitApplication(bool allow_confirm);

src/duckstation-qt/qthost.cpp

@@ -743,6 +743,9 @@ void EmuThread::startFullscreenUI()
     Host::ReportErrorAsync("Error", error.GetDescription());
     return;
   }
+
+  m_is_fullscreen_ui_started = true;
+  emit fullscreenUIStartedOrStopped(true);
 }
 
 void EmuThread::stopFullscreenUI()
@@ -758,7 +761,10 @@ void EmuThread::stopFullscreenUI()
   }
 
   if (m_is_fullscreen_ui_started)
+  {
     GPUThread::StopFullscreenUI();
+    emit fullscreenUIStartedOrStopped(false);
+  }
 }
 
 void EmuThread::bootSystem(std::shared_ptr<SystemBootParameters> params)
@@ -1023,11 +1029,6 @@ void Host::OnSystemDestroyed()
   emit g_emu_thread->systemDestroyed();
 }
 
-void Host::OnFullscreenUIStartedOrStopped(bool started)
-{
-  g_emu_thread->setFullscreenUIStarted(started);
-}
-
 void Host::OnGPUThreadRunIdleChanged(bool is_active)
 {
   g_emu_thread->setGPUThreadRunIdle(is_active);
@@ -1823,15 +1824,6 @@ void EmuThread::setGPUThreadRunIdle(bool active)
   g_emu_thread->startBackgroundControllerPollTimer();
 }
 
-void EmuThread::setFullscreenUIStarted(bool started)
-{
-  if (m_is_fullscreen_ui_started == started)
-    return;
-
-  m_is_fullscreen_ui_started = started;
-  emit fullscreenUIStartedOrStopped(started);
-}
-
 void EmuThread::start()
 {
   AssertMsg(!g_emu_thread, "Emu thread does not exist");

src/duckstation-qt/qthost.h

@@ -103,7 +103,6 @@ public:
 
   void startBackgroundControllerPollTimer();
   void stopBackgroundControllerPollTimer();
-  void setFullscreenUIStarted(bool started);
   void wakeThread();
 
   bool shouldRenderToMain() const;

src/duckstation-regtest/regtest_host.cpp

@@ -296,11 +296,6 @@ void Host::OnGPUThreadRunIdleChanged(bool is_active)
   //
 }
 
-void Host::OnFullscreenUIStartedOrStopped(bool started)
-{
-  //
-}
-
 void Host::OnPerformanceCountersUpdated(const GPUBackend* gpu_backend)
 {
   //