### 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)

ERR_FAIL_COND_V_MSG(!Thread::is_main_thread(), Ref<MultiplayerAPI>(), "Multiplayer can only be manipulated from the main thread.");
	Ref<MultiplayerAPI> out = multiplayer;
	for (const KeyValue<NodePath, Ref<MultiplayerAPI>> &E : custom_multiplayers) {
		const Vector<StringName> snames = E.key.get_names();
		const Vector<StringName> tnames = p_for_path.get_names();
		if (tnames.size() < snames.size()) {
			continue;
		}
		const StringName *sptr = snames.ptr();
		const StringName *nptr = tnames.ptr();
		bool valid = true;
		for (int i = 0; i < snames.size(); i++) {
			if (sptr[i] != nptr[i]) {
				valid = false;
				break;
			}
		}
		if (valid) {
			out = E.value;
			break;
		}
	}
	return out;
}
void SceneTree::set_multiplayer(Ref<MultiplayerAPI> p_multiplayer, const NodePath &p_root_path) {