### 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)

if (p_node->is_class("CanvasItem") || p_node->is_class("Node3D")) {
		Color color(1, 1, 1, 1);
		bool visible_on_screen = p_node->call("is_visible_in_tree");
		if (!visible_on_screen) {
			color.a = 0.6;
		}
		int idx = p_item->get_button_by_id(0, BUTTON_VISIBILITY);
		p_item->set_button_color(0, idx, color);
	}
}
void SceneTreeEditor::_node_script_changed(Node *p_node) {
	if (tree_dirty) {
		return;
	}
	MessageQueue::get_singleton()->push_call(this, "_update_tree");
	tree_dirty = true;
}
void SceneTreeEditor::_node_removed(Node *p_node) {