### 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)

root_lock++;
	MainLoop::process(p_time);
	process_time = p_time;
	if (multiplayer_poll) {
		multiplayer->poll();
		for (KeyValue<NodePath, Ref<MultiplayerAPI>> &E : custom_multiplayers) {
			E.value->poll();
		}
	}
	emit_signal(SNAME("process_frame"));
	MessageQueue::get_singleton()->flush(); 
	flush_transform_notifications();
	_process(false);
	_flush_ugc();
	MessageQueue::get_singleton()->flush(); 
	flush_transform_notifications(); 
	root_lock--;
	_flush_delete_queue();
	process_timers(p_time, false); 
	process_tweens(p_time, false);
	flush_transform_notifications();