### 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)

if (!p_parent) {
		p_parent = tree->get_root();
		filter_term_warning.clear();
	}
	if (!p_parent) {
		// Tree is empty, nothing to do here.
		return false;
	}
	bool keep_for_children = false;
	for (TreeItem *child = p_parent->get_first_child(); child; child = child->get_next()) {
		// Always keep if at least one of the children are kept.
		keep_for_children = _update_filter(child, p_scroll_to_selected) || keep_for_children;
	}
	// Now find other reasons to keep this Node, too.
	PackedStringArray terms = filter.to_lower().split_spaces();
	bool keep = _item_matches_all_terms(p_parent, terms);
	bool selectable = keep;
	if (keep && !valid_types.is_empty()) {
		selectable = false;
		Node *n = get_node(p_parent->get_metadata(0));
		for (const StringName &E : valid_types) {
			if (n->is_class(E) ||
					EditorNode::get_singleton()->is_object_of_custom_type(n, E)) {
				selectable = true;
				break;
			}
		}
	}