### 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)

_flush_delete_queue();
	_flush_ugc();
	initialized = false;
	MainLoop::finalize();
	if (root) {
		root->_set_tree(nullptr);
		root->_propagate_after_exit_tree();
		memdelete(root); 
		root = nullptr;
	}
	
	
	_flush_delete_queue();
	
	for (Ref<SceneTreeTimer> &timer : timers) {
		timer->release_connections();
	}
	timers.clear();
	
	for (Ref<Tween> &tween : tweens) {
		tween->clear();
	}
	tweens.clear();
}
void SceneTree::quit(int p_exit_code) {