### 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)

switch (p_notification) {
		case NOTIFICATION_TRANSLATION_CHANGED: {
			if (!Engine::get_singleton()->is_editor_hint()) {
				get_root()->propagate_notification(p_notification);
			}
		} break;
		case NOTIFICATION_OS_MEMORY_WARNING:
		case NOTIFICATION_OS_IME_UPDATE:
		case NOTIFICATION_WM_ABOUT:
		case NOTIFICATION_CRASH:
		case NOTIFICATION_APPLICATION_RESUMED:
		case NOTIFICATION_APPLICATION_PAUSED:
		case NOTIFICATION_APPLICATION_FOCUS_IN:
		case NOTIFICATION_APPLICATION_FOCUS_OUT: {
			
			get_root()->propagate_notification(p_notification);
		} break;
	}
}
bool SceneTree::is_auto_accept_quit() const {
	return accept_quit;
}
void SceneTree::set_auto_accept_quit(bool p_enable) {
	accept_quit = p_enable;
}
bool SceneTree::is_quit_on_go_back() const {
	return quit_on_go_back;
}
void SceneTree::set_quit_on_go_back(bool p_enable) {