### 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)

}
				n->_call_shortcut_input(p_input);
				break;
			}
			case CALL_INPUT_TYPE_UNHANDLED_INPUT:
				n->_call_unhandled_input(p_input);
				break;
			case CALL_INPUT_TYPE_UNHANDLED_KEY_INPUT:
				n->_call_unhandled_key_input(p_input);
				break;
		}
	}
	for (const ObjectID &id : no_context_node_ids) {
		if (p_viewport->is_input_handled()) {
			break;
		}
		Node *n = Object::cast_to<Node>(ObjectDB::get_instance(id));
		if (n) {
			n->_call_shortcut_input(p_input);
		}
	}
	{
		_THREAD_SAFE_METHOD_
		nodes_removed_on_group_call_lock--;
		if (nodes_removed_on_group_call_lock == 0) {
			nodes_removed_on_group_call.clear();
		}
	}
}
void SceneTree::_call_group_flags(const Variant **p_args, int p_argcount, Callable::CallError &r_error) {