clutter/context: Create a StageManager when constructed

Instead of delegating that to StageManager.get_default and add a private getter
for it.

Part-of: <https://gitlab.gnome.org/GNOME/mutter/-/merge_requests/3962>

clutter/clutter/clutter-context-private.h

@@ -20,6 +20,7 @@
 #pragma once
 
 #include "clutter/clutter-context.h"
+#include "clutter-stage-manager-private.h"
 
 struct _ClutterContext
 {
@@ -45,3 +46,5 @@ struct _ClutterContext
 
   gboolean show_fps;
 };
+
+ClutterStageManager * clutter_context_get_stage_manager (ClutterContext *context);