### 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)

item->set_custom_color(0, get_theme_color(SNAME("disabled_font_color"), SNAME("Editor")));
			item->set_selectable(0, false);
		}
	}
}
void SceneTreeEditor::_node_visibility_changed(Node *p_node) {
	if (!p_node || (p_node != get_scene_node() && !p_node->get_owner())) {
		return;
	}
	TreeItem *item = _find(tree->get_root(), p_node->get_path());
	if (!item) {
		return;
	}
	int idx = item->get_button_by_id(0, BUTTON_VISIBILITY);
	ERR_FAIL_COND(idx == -1);
	bool node_visible = false;
	if (p_node->is_class("CanvasItem") || p_node->is_class("CanvasLayer") || p_node->is_class("Window")) {
		node_visible = p_node->call("is_visible");
		CanvasItemEditor::get_singleton()->get_viewport_control()->queue_redraw();
	} else if (p_node->is_class("Node3D")) {
		node_visible = p_node->call("is_visible");
	}
	if (node_visible) {
		item->set_button(0, idx, get_theme_icon(SNAME("GuiVisibilityVisible"), SNAME("EditorIcons")));
	} else {
		item->set_button(0, idx, get_theme_icon(SNAME("GuiVisibilityHidden"), SNAME("EditorIcons")));
	}
	_update_visibility_color(p_node, item);
}
void SceneTreeEditor::_update_visibility_color(Node *p_node, TreeItem *p_item) {