### Godot version

v4.1.beta1.mono.official [828ec2c5d]

### System information

Godot v4.1.beta1.mono - Windows 10.0.22621 - Vulkan (Forward+) - dedicated NVIDIA GeForce RTX 4070 Ti (NVIDIA; 31.0.15.3598) - 13th Gen Intel(R) Core(TM) i5-13600K (20 Threads)

### Issue description

After the recent refactoring (https://github.com/godotengine/godot/pull/75901), Node process priority seems to be unimplemented. `ComparatorWithPriority` is no longer used anywhere in the engine code.

Nodes behave as if they all had the same priority, and running with `--single-threaded-scene` makes no difference.

### Steps to reproduce

Run the MRP, look at the order in which the nodes print their names.

In Godot 4.0.3: First, Default, Last
In Godot 4.1-beta1: Default, First, Last

### Minimal reproduction project

[ProcessPriorityBug.zip](https://github.com/godotengine/godot/files/11693028/ProcessPriorityBug.zip)

ERR_FAIL_COND_MSG(!Thread::is_main_thread(), "Changing scene can only be done from the main thread.");
	if (current_scene) {
		memdelete(current_scene);
		current_scene = nullptr;
	}
	
	if (unlikely(_quit)) {
		if (p_to) { 
			memdelete(p_to);
		}
		return;
	}
	if (p_to) {
		current_scene = p_to;
		root->add_child(p_to);
		
		root->update_mouse_cursor_state();
	}
}
Error SceneTree::change_scene_to_file(const String &p_path) {
	ERR_FAIL_COND_V_MSG(!Thread::is_main_thread(), ERR_INVALID_PARAMETER, "Changing scene can only be done from the main thread.");
	Ref<PackedScene> new_scene = ResourceLoader::load(p_path);
	if (new_scene.is_null()) {
		return ERR_CANT_OPEN;
	}
	return change_scene_to_packed(new_scene);
}
Error SceneTree::change_scene_to_packed(const Ref<PackedScene> &p_scene) {