### 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)

}
		} else {
			error->set_text(text);
			error->set_meta("invalid_character", true);
			error->set_meta("same_unique_name", false);
			error->popup_centered();
		}
	}
	// Trim leading/trailing whitespace to prevent node names from containing accidental whitespace, which would make it more difficult to get the node via `get_node()`.
	new_name = new_name.strip_edges();
	if (new_name.is_empty()) {
		// If name is empty, fallback to class name.
		if (GLOBAL_GET("editor/naming/node_name_casing").operator int() != NAME_CASING_PASCAL_CASE) {
			new_name = Node::adjust_name_casing(p_node->get_class());
		} else {
			new_name = p_node->get_class();
		}
	}
	if (new_name == p_node->get_name()) {
		if (item->get_text(0).is_empty()) {
			item->set_text(0, new_name);
		}
		return;
	}
	// We previously made sure name is not the same as current name so that it won't complain about already used unique name when not changing name.
	if (p_node->is_unique_name_in_owner() && get_tree()->get_edited_scene_root()->get_node_or_null("%" + new_name)) {
		String text = TTR("Another node already uses this unique name in the scene.");
		if (error->is_visible()) {
			if (!error->get_meta("same_unique_name", false)) {
				error->set_text(error->get_text() + "\n\n" + text);