### 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)

bool is_pinned = AnimationPlayerEditor::get_singleton()->get_player() == p_node && AnimationPlayerEditor::get_singleton()->is_pinned();
			if (is_pinned) {
				item->add_button(0, get_theme_icon(SNAME("Pin"), SNAME("EditorIcons")), BUTTON_PIN, false, TTR("AnimationPlayer is pinned.\nClick to unpin."));
			}
		}
	}
	if (editor_selection) {
		if (editor_selection->is_selected(p_node)) {
			item->select(0);
		}
	}
	if (selected == p_node) {
		if (!editor_selection) {
			item->select(0);
		}
		item->set_as_cursor(0);
	}
	for (int i = 0; i < p_node->get_child_count(); i++) {
		_add_nodes(p_node->get_child(i), item);
	}
	if (valid_types.size()) {
		bool valid = false;
		for (const StringName &E : valid_types) {
			if (p_node->is_class(E) ||
					EditorNode::get_singleton()->is_object_of_custom_type(p_node, E)) {
				valid = true;
				break;
			}
		}
		if (!valid) {