### 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_V_MSG(p_scene.is_null(), ERR_INVALID_PARAMETER, "Can't change to a null scene. Use unload_current_scene() if you wish to unload it.");
	Node *new_scene = p_scene->instantiate();
	ERR_FAIL_COND_V(!new_scene, ERR_CANT_CREATE);
	call_deferred(SNAME("_change_scene"), new_scene);
	return OK;
}
Error SceneTree::reload_current_scene() {
	ERR_FAIL_COND_V_MSG(!Thread::is_main_thread(), ERR_INVALID_PARAMETER, "Reloading scene can only be done from the main thread.");
	ERR_FAIL_COND_V(!current_scene, ERR_UNCONFIGURED);
	String fname = current_scene->get_scene_file_path();
	return change_scene_to_file(fname);
}
void SceneTree::unload_current_scene() {
	ERR_FAIL_COND_MSG(!Thread::is_main_thread(), "Unloading the current scene can only be done from the main thread.");
	if (current_scene) {
		memdelete(current_scene);
		current_scene = nullptr;
	}
}
void SceneTree::add_current_scene(Node *p_current) {
	ERR_FAIL_COND_MSG(!Thread::is_main_thread(), "Adding a current scene can only be done from the main thread.");
	current_scene = p_current;
	root->add_child(p_current);
}
Ref<SceneTreeTimer> SceneTree::create_timer(double p_delay_sec, bool p_process_always, bool p_process_in_physics, bool p_ignore_time_scale) {