### 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)

_THREAD_SAFE_METHOD_
	Ref<SceneTreeTimer> stt;
	stt.instantiate();
	stt->set_process_always(p_process_always);
	stt->set_time_left(p_delay_sec);
	stt->set_process_in_physics(p_process_in_physics);
	stt->set_ignore_time_scale(p_ignore_time_scale);
	timers.push_back(stt);
	return stt;
}
Ref<Tween> SceneTree::create_tween() {
	_THREAD_SAFE_METHOD_
	Ref<Tween> tween = memnew(Tween(true));
	tweens.push_back(tween);
	return tween;
}
TypedArray<Tween> SceneTree::get_processed_tweens() {
	_THREAD_SAFE_METHOD_
	TypedArray<Tween> ret;
	ret.resize(tweens.size());
	int i = 0;
	for (const Ref<Tween> &tween : tweens) {
		ret[i] = tween;
		i++;
	}
	return ret;
}
Ref<MultiplayerAPI> SceneTree::get_multiplayer(const NodePath &p_for_path) const {