### 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)

HBoxContainer *filter_hbc = memnew(HBoxContainer);
	content->add_child(filter_hbc);
	filter = memnew(LineEdit);
	filter->set_h_size_flags(Control::SIZE_EXPAND_FILL);
	filter->set_placeholder(TTR("Filter Nodes"));
	filter->set_clear_button_enabled(true);
	filter->add_theme_constant_override("minimum_character_width", 0);
	filter->connect("text_changed", callable_mp(this, &SceneTreeDialog::_filter_changed));
	filter_hbc->add_child(filter);
	// Add 'Show All' button to HBoxContainer next to the filter, visible only when valid_types is defined.
	show_all_nodes = memnew(CheckButton);
	show_all_nodes->set_text(TTR("Show All"));
	show_all_nodes->connect("toggled", callable_mp(this, &SceneTreeDialog::_show_all_nodes_changed));
	show_all_nodes->set_h_size_flags(Control::SIZE_SHRINK_BEGIN);
	show_all_nodes->hide();
	filter_hbc->add_child(show_all_nodes);
	tree = memnew(SceneTreeEditor(false, false, true));
	tree->set_v_size_flags(Control::SIZE_EXPAND_FILL);
	tree->get_scene_tree()->connect("item_activated", callable_mp(this, &SceneTreeDialog::_select));
	// Initialize button state, must be done after the tree has been created to update its 'show_all_nodes' flag.
	// This is also done before adding the tree to the content to avoid triggering unnecessary tree filtering.
	show_all_nodes->set_pressed(EditorSettings::get_singleton()->get_project_metadata("editor_metadata", "show_all_nodes_for_node_selection", false));
	content->add_child(tree);
	// Disable the OK button when no node is selected.
	get_ok_button()->set_disabled(!tree->get_selected());
	tree->connect("node_selected", callable_mp(this, &SceneTreeDialog::_selected_changed));
}
SceneTreeDialog::~SceneTreeDialog() {
}