System: Set GPU timing enable on init/destroy

It's really unrelated to the guest GPU.

src/core/gpu.cpp

@@ -80,9 +80,6 @@ GPU::~GPU()
   JoinScreenshotThreads();
   DestroyDeinterlaceTextures();
   g_gpu_device->RecycleTexture(std::move(m_chroma_smoothing_texture));
-
-  if (g_gpu_device)
-    g_gpu_device->SetGPUTimingEnabled(false);
 }
 
 bool GPU::Initialize()
@@ -101,8 +98,6 @@ bool GPU::Initialize()
     return false;
   }
 
-  g_gpu_device->SetGPUTimingEnabled(g_settings.display_show_gpu_usage);
-
 #ifdef PSX_GPU_STATS
   s_active_gpu_cycles = 0;
   s_active_gpu_cycles_frames = 0;
@@ -147,8 +142,6 @@ void GPU::UpdateSettings(const Settings& old_settings)
       Panic("Failed to compile display pipeline on settings change.");
     }
   }
-
-  g_gpu_device->SetGPUTimingEnabled(g_settings.display_show_gpu_usage);
 }
 
 void GPU::CPUClockChanged()

src/core/system.cpp

@@ -1911,6 +1911,9 @@ bool System::Initialize(bool force_software_renderer, Error* error)
 
   s_cpu_thread_handle = Threading::ThreadHandle::GetForCallingThread();
 
+  if (g_settings.display_show_gpu_usage)
+    g_gpu_device->SetGPUTimingEnabled(true);
+
   UpdateThrottlePeriod();
   UpdateMemorySaveStateSettings();
   return true;
@@ -1971,6 +1974,7 @@ void System::DestroySystem()
   // Restore present-all-frames behavior.
   if (s_keep_gpu_device_on_shutdown && g_gpu_device)
   {
+    g_gpu_device->SetGPUTimingEnabled(false);
     UpdateDisplayVSync();
   }
   else
@@ -4447,6 +4451,9 @@ void System::CheckForSettingsChanges(const Settings& old_settings)
       UpdateSpeedLimiterState();
     }
 
+    if (g_settings.display_show_gpu_usage != old_settings.display_show_gpu_usage)
+      g_gpu_device->SetGPUTimingEnabled(g_settings.display_show_gpu_usage);
+
     if (g_settings.inhibit_screensaver != old_settings.inhibit_screensaver)
     {
       if (g_settings.inhibit_screensaver)