### 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_
	OS::get_singleton()->set_exit_code(p_exit_code);
	_quit = true;
}
void SceneTree::_main_window_close() {
	if (accept_quit) {
		_quit = true;
	}
}
void SceneTree::_main_window_go_back() {
	if (quit_on_go_back) {
		_quit = true;
	}
}
void SceneTree::_main_window_focus_in() {
	Input *id = Input::get_singleton();
	if (id) {
		id->ensure_touch_mouse_raised();
	}
}
void SceneTree::_notification(int p_notification) {