### 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)

}
	{
		// Display the node name in all tooltips so that long node names can be previewed
		// without having to rename them.
		String tooltip = String(p_node->get_name());
		if (p_node == get_scene_node() && p_node->get_scene_inherited_state().is_valid()) {
			item->add_button(0, get_theme_icon(SNAME("InstanceOptions"), SNAME("EditorIcons")), BUTTON_SUBSCENE, false, TTR("Open in Editor"));
			tooltip += String("\n" + TTR("Inherits:") + " " + p_node->get_scene_inherited_state()->get_path());
		} else if (p_node != get_scene_node() && !p_node->get_scene_file_path().is_empty() && can_open_instance) {
			item->add_button(0, get_theme_icon(SNAME("InstanceOptions"), SNAME("EditorIcons")), BUTTON_SUBSCENE, false, TTR("Open in Editor"));
			tooltip += String("\n" + TTR("Instance:") + " " + p_node->get_scene_file_path());
		}
		StringName custom_type = EditorNode::get_singleton()->get_object_custom_type_name(p_node);
		tooltip += String("\n" + TTR("Type:") + " " + (custom_type != StringName() ? String(custom_type) : p_node->get_class()));
		if (!p_node->get_editor_description().is_empty()) {
			const PackedInt32Array boundaries = TS->string_get_word_breaks(p_node->get_editor_description(), "", 80);
			tooltip += "\n";
			for (int i = 0; i < boundaries.size(); i += 2) {
				const int start = boundaries[i];
				const int end = boundaries[i + 1];
				tooltip += "\n" + p_node->get_editor_description().substr(start, end - start + 1).rstrip("\n");
			}
		}
		item->set_tooltip_text(0, tooltip);
	}
	if (can_open_instance && is_scene_tree_dock) { // Show buttons only when necessary (SceneTreeDock) to avoid crashes.
		if (!p_node->is_connected("script_changed", callable_mp(this, &SceneTreeEditor::_node_script_changed))) {
			p_node->connect("script_changed", callable_mp(this, &SceneTreeEditor::_node_script_changed).bind(p_node));
		}
		Ref<Script> scr = p_node->get_script();