### 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)

MethodInfo mi;
	mi.name = "call_group_flags";
	mi.arguments.push_back(PropertyInfo(Variant::INT, "flags"));
	mi.arguments.push_back(PropertyInfo(Variant::STRING_NAME, "group"));
	mi.arguments.push_back(PropertyInfo(Variant::STRING_NAME, "method"));
	ClassDB::bind_vararg_method(METHOD_FLAGS_DEFAULT, "call_group_flags", &SceneTree::_call_group_flags, mi);
	ClassDB::bind_method(D_METHOD("notify_group_flags", "call_flags", "group", "notification"), &SceneTree::notify_group_flags);
	ClassDB::bind_method(D_METHOD("set_group_flags", "call_flags", "group", "property", "value"), &SceneTree::set_group_flags);
	MethodInfo mi2;
	mi2.name = "call_group";
	mi2.arguments.push_back(PropertyInfo(Variant::STRING_NAME, "group"));
	mi2.arguments.push_back(PropertyInfo(Variant::STRING_NAME, "method"));
	ClassDB::bind_vararg_method(METHOD_FLAGS_DEFAULT, "call_group", &SceneTree::_call_group, mi2);
	ClassDB::bind_method(D_METHOD("notify_group", "group", "notification"), &SceneTree::notify_group);
	ClassDB::bind_method(D_METHOD("set_group", "group", "property", "value"), &SceneTree::set_group);
	ClassDB::bind_method(D_METHOD("get_nodes_in_group", "group"), &SceneTree::_get_nodes_in_group);
	ClassDB::bind_method(D_METHOD("get_first_node_in_group", "group"), &SceneTree::get_first_node_in_group);
	ClassDB::bind_method(D_METHOD("set_current_scene", "child_node"), &SceneTree::set_current_scene);
	ClassDB::bind_method(D_METHOD("get_current_scene"), &SceneTree::get_current_scene);
	ClassDB::bind_method(D_METHOD("change_scene_to_file", "path"), &SceneTree::change_scene_to_file);
	ClassDB::bind_method(D_METHOD("change_scene_to_packed", "packed_scene"), &SceneTree::change_scene_to_packed);
	ClassDB::bind_method(D_METHOD("reload_current_scene"), &SceneTree::reload_current_scene);
	ClassDB::bind_method(D_METHOD("unload_current_scene"), &SceneTree::unload_current_scene);
	ClassDB::bind_method(D_METHOD("_change_scene"), &SceneTree::_change_scene);
	ClassDB::bind_method(D_METHOD("set_multiplayer", "multiplayer", "root_path"), &SceneTree::set_multiplayer, DEFVAL(NodePath()));
	ClassDB::bind_method(D_METHOD("get_multiplayer", "for_path"), &SceneTree::get_multiplayer, DEFVAL(NodePath()));
	ClassDB::bind_method(D_METHOD("set_multiplayer_poll_enabled", "enabled"), &SceneTree::set_multiplayer_poll_enabled);
	ClassDB::bind_method(D_METHOD("is_multiplayer_poll_enabled"), &SceneTree::is_multiplayer_poll_enabled);
	ADD_PROPERTY(PropertyInfo(Variant::BOOL, "auto_accept_quit"), "set_auto_accept_quit", "is_auto_accept_quit");
	ADD_PROPERTY(PropertyInfo(Variant::BOOL, "quit_on_go_back"), "set_quit_on_go_back", "is_quit_on_go_back");