### 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.");
	multiplayer_poll = p_enabled;
}
bool SceneTree::is_multiplayer_poll_enabled() const {
	return multiplayer_poll;
}
void SceneTree::_bind_methods() {
	ClassDB::bind_method(D_METHOD("get_root"), &SceneTree::get_root);
	ClassDB::bind_method(D_METHOD("has_group", "name"), &SceneTree::has_group);
	ClassDB::bind_method(D_METHOD("is_auto_accept_quit"), &SceneTree::is_auto_accept_quit);
	ClassDB::bind_method(D_METHOD("set_auto_accept_quit", "enabled"), &SceneTree::set_auto_accept_quit);
	ClassDB::bind_method(D_METHOD("is_quit_on_go_back"), &SceneTree::is_quit_on_go_back);
	ClassDB::bind_method(D_METHOD("set_quit_on_go_back", "enabled"), &SceneTree::set_quit_on_go_back);
	ClassDB::bind_method(D_METHOD("set_debug_collisions_hint", "enable"), &SceneTree::set_debug_collisions_hint);
	ClassDB::bind_method(D_METHOD("is_debugging_collisions_hint"), &SceneTree::is_debugging_collisions_hint);
	ClassDB::bind_method(D_METHOD("set_debug_paths_hint", "enable"), &SceneTree::set_debug_paths_hint);
	ClassDB::bind_method(D_METHOD("is_debugging_paths_hint"), &SceneTree::is_debugging_paths_hint);
	ClassDB::bind_method(D_METHOD("set_debug_navigation_hint", "enable"), &SceneTree::set_debug_navigation_hint);
	ClassDB::bind_method(D_METHOD("is_debugging_navigation_hint"), &SceneTree::is_debugging_navigation_hint);
	ClassDB::bind_method(D_METHOD("set_edited_scene_root", "scene"), &SceneTree::set_edited_scene_root);
	ClassDB::bind_method(D_METHOD("get_edited_scene_root"), &SceneTree::get_edited_scene_root);
	ClassDB::bind_method(D_METHOD("set_pause", "enable"), &SceneTree::set_pause);
	ClassDB::bind_method(D_METHOD("is_paused"), &SceneTree::is_paused);
	ClassDB::bind_method(D_METHOD("create_timer", "time_sec", "process_always", "process_in_physics", "ignore_time_scale"), &SceneTree::create_timer, DEFVAL(true), DEFVAL(false), DEFVAL(false));
	ClassDB::bind_method(D_METHOD("create_tween"), &SceneTree::create_tween);
	ClassDB::bind_method(D_METHOD("get_processed_tweens"), &SceneTree::get_processed_tweens);
	ClassDB::bind_method(D_METHOD("get_node_count"), &SceneTree::get_node_count);
	ClassDB::bind_method(D_METHOD("get_frame"), &SceneTree::get_frame);
	ClassDB::bind_method(D_METHOD("quit", "exit_code"), &SceneTree::quit, DEFVAL(EXIT_SUCCESS));
	ClassDB::bind_method(D_METHOD("queue_delete", "obj"), &SceneTree::queue_delete);