### 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(), "Pause can only be set from the main thread.");
	if (p_enabled == paused) {
		return;
	}
	paused = p_enabled;
	PhysicsServer3D::get_singleton()->set_active(!p_enabled);
	PhysicsServer2D::get_singleton()->set_active(!p_enabled);
	if (get_root()) {
		get_root()->_propagate_pause_notification(p_enabled);
	}
}
bool SceneTree::is_paused() const {
	return paused;
}
void SceneTree::_process_group(ProcessGroup *p_group, bool p_physics) {