### Godot version

4.1???- 4.1.beta1 - 4.1.beta3 - 030c1a950e8cee19e72104b4fb79caba17a2d678

### System information

Tested only Linux versions.

### Issue description

**Regression from 4.0.3.stable:**
In the editor UI set the  `Node->process_mode` to `Disabled` or `When Paused`-> in the scene editor the subtree is grayed out. Save the scene and close the editor. 
It doesn*t need to be the current scene. 
Start it again, the subtree in the scene is no longer gray. The *.tscn file is updated correctly, e.g.:
```
  [node name="Node2D" type="Node2D"]
+process_mode = 4
```
 Workaround is to toggle the process mode again,
 close and open or reload the scene,
(or disable the `interface/scene_tabs/restore_scenes_on_load` option ...)

### Image gallery
![image](https://github.com/godotengine/godot/assets/4047289/b6a90b34-c9ec-4380-b953-f1e64cf72934)
![image](https://github.com/godotengine/godot/assets/4047289/978d4ab1-553b-49e6-bb76-6b18c22a942b)
![image](https://github.com/godotengine/godot/assets/4047289/4254f160-75d5-4186-9802-53b487d8d2f3)

ERR_FAIL_COND_V(!is_inside_tree(), nullptr);
	return get_tree()->get_edited_scene_root();
}
void SceneTreeEditor::_cell_button_pressed(Object *p_item, int p_column, int p_id, MouseButton p_button) {
	if (p_button != MouseButton::LEFT) {
		return;
	}
	if (connect_to_script_mode) {
		return; 
	}
	TreeItem *item = Object::cast_to<TreeItem>(p_item);
	ERR_FAIL_COND(!item);
	NodePath np = item->get_metadata(0);
	Node *n = get_node(np);
	ERR_FAIL_COND(!n);
	EditorUndoRedoManager *undo_redo = EditorUndoRedoManager::get_singleton();
	if (p_id == BUTTON_SUBSCENE) {
		if (n == get_scene_node()) {
			if (n && n->get_scene_inherited_state().is_valid()) {
				emit_signal(SNAME("open"), n->get_scene_inherited_state()->get_path());
			}
		} else {
			emit_signal(SNAME("open"), n->get_scene_file_path());
		}
	} else if (p_id == BUTTON_SCRIPT) {
		Ref<Script> script_typed = n->get_script();