### 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(idle_callback_count >= MAX_IDLE_CALLBACKS);
	idle_callbacks[idle_callback_count++] = p_callback;
}
void SceneTree::get_argument_options(const StringName &p_function, int p_idx, List<String> *r_options) const {
	if (p_function == "change_scene_to_file") {
		Ref<DirAccess> dir_access = DirAccess::create(DirAccess::ACCESS_RESOURCES);
		List<String> directories;
		directories.push_back(dir_access->get_current_dir());
		while (!directories.is_empty()) {
			dir_access->change_dir(directories.back()->get());
			directories.pop_back();
			dir_access->list_dir_begin();
			String filename = dir_access->get_next();
			while (!filename.is_empty()) {
				if (filename == "." || filename == "..") {
					filename = dir_access->get_next();
					continue;
				}
				if (dir_access->dir_exists(filename)) {
					directories.push_back(dir_access->get_current_dir().path_join(filename));
				} else if (filename.ends_with(".tscn") || filename.ends_with(".scn")) {
					r_options->push_back("\"" + dir_access->get_current_dir().path_join(filename) + "\"");
				}
				filename = dir_access->get_next();
			}
		}
	}
}
void SceneTree::set_disable_node_threading(bool p_disable) {