### 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)

if (!(p_call_flags & GROUP_CALL_DEFERRED)) {
				gr_nodes[i]->set(p_name, p_value);
			} else {
				MessageQueue::get_singleton()->push_set(gr_nodes[i], p_name, p_value);
			}
		}
	} else {
		for (int i = 0; i < gr_node_count; i++) {
			if (nodes_removed_on_group_call.has(gr_nodes[i])) {
				continue;
			}
			if (!(p_call_flags & GROUP_CALL_DEFERRED)) {
				gr_nodes[i]->set(p_name, p_value);
			} else {
				MessageQueue::get_singleton()->push_set(gr_nodes[i], p_name, p_value);
			}
		}
	}
	{
		_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::notify_group(const StringName &p_group, int p_notification) {
	notify_group_flags(GROUP_CALL_DEFAULT, p_group, p_notification);
}
void SceneTree::set_group(const StringName &p_group, const String &p_name, const Variant &p_value) {