### 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_MSG(!Thread::is_main_thread(), "Multiplayer can only be manipulated from the main thread.");
	if (p_root_path.is_empty()) {
		ERR_FAIL_COND(!p_multiplayer.is_valid());
		if (multiplayer.is_valid()) {
			multiplayer->object_configuration_remove(nullptr, NodePath("/" + root->get_name()));
		}
		multiplayer = p_multiplayer;
		multiplayer->object_configuration_add(nullptr, NodePath("/" + root->get_name()));
	} else {
		if (custom_multiplayers.has(p_root_path)) {
			custom_multiplayers[p_root_path]->object_configuration_remove(nullptr, p_root_path);
		}
		if (p_multiplayer.is_valid()) {
			custom_multiplayers[p_root_path] = p_multiplayer;
			p_multiplayer->object_configuration_add(nullptr, p_root_path);
		}
	}
}
void SceneTree::set_multiplayer_poll_enabled(bool p_enabled) {