### 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)

#include "core/io/image_loader.h"
#include "core/io/marshalls.h"
#include "core/io/resource_loader.h"
#include "core/object/message_queue.h"
#include "core/object/worker_thread_pool.h"
#include "core/os/keyboard.h"
#include "core/os/os.h"
#include "core/string/print_string.h"
#include "node.h"
#include "scene/animation/tween.h"
#include "scene/debugger/scene_debugger.h"
#include "scene/gui/control.h"
#include "scene/main/multiplayer_api.h"
#include "scene/main/viewport.h"
#include "scene/resources/environment.h"
#include "scene/resources/font.h"
#include "scene/resources/material.h"
#include "scene/resources/mesh.h"
#include "scene/resources/packed_scene.h"
#include "scene/resources/world_2d.h"
#include "scene/resources/world_3d.h"
#include "scene/scene_string_names.h"
#include "servers/display_server.h"
#include "servers/navigation_server_3d.h"
#include "servers/physics_server_2d.h"
#include "servers/physics_server_3d.h"
#include "window.h"
#include <stdio.h>
#include <stdlib.h>
void SceneTreeTimer::_bind_methods() {