### 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)

editor_selection->add_node(n);
		set_selected(n);
		TabContainer *tab_container = Object::cast_to<TabContainer>(NodeDock::get_singleton()->get_parent());
		NodeDock::get_singleton()->get_parent()->call("set_current_tab", tab_container->get_tab_idx_from_control(NodeDock::get_singleton()));
		NodeDock::get_singleton()->show_connections();
	} else if (p_id == BUTTON_GROUPS) {
		editor_selection->clear();
		editor_selection->add_node(n);
		set_selected(n);
		TabContainer *tab_container = Object::cast_to<TabContainer>(NodeDock::get_singleton()->get_parent());
		NodeDock::get_singleton()->get_parent()->call("set_current_tab", tab_container->get_tab_idx_from_control(NodeDock::get_singleton()));
		NodeDock::get_singleton()->show_groups();
	} else if (p_id == BUTTON_UNIQUE) {
		undo_redo->create_action(TTR("Disable Scene Unique Name"));
		undo_redo->add_do_method(n, "set_unique_name_in_owner", false);
		undo_redo->add_undo_method(n, "set_unique_name_in_owner", true);
		undo_redo->add_do_method(this, "_update_tree");
		undo_redo->add_undo_method(this, "_update_tree");
		undo_redo->commit_action();
	}
}
void SceneTreeEditor::_toggle_visible(Node *p_node) {
	if (p_node->has_method("is_visible") && p_node->has_method("set_visible")) {
		bool v = bool(p_node->call("is_visible"));
		EditorUndoRedoManager *undo_redo = EditorUndoRedoManager::get_singleton();
		undo_redo->add_do_method(p_node, "set_visible", !v);
		undo_redo->add_undo_method(p_node, "set_visible", v);
	}
}
void SceneTreeEditor::_add_nodes(Node *p_node, TreeItem *p_parent) {