### 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)

root->set_sdf_scale(sdf_scale);
#ifndef _3D_DISABLED
	{ // Load default fallback environment.
		// Get possible extensions.
		List<String> exts;
		ResourceLoader::get_recognized_extensions_for_type("Environment", &exts);
		String ext_hint;
		for (const String &E : exts) {
			if (!ext_hint.is_empty()) {
				ext_hint += ",";
			}
			ext_hint += "*." + E;
		}
		// Get path.
		String env_path = GLOBAL_DEF(PropertyInfo(Variant::STRING, "rendering/environment/defaults/default_environment", PROPERTY_HINT_FILE, ext_hint), "");
		// Setup property.
		env_path = env_path.strip_edges();
		if (!env_path.is_empty()) {
			Ref<Environment> env = ResourceLoader::load(env_path);
			if (env.is_valid()) {
				root->get_world_3d()->set_fallback_environment(env);
			} else {
				if (Engine::get_singleton()->is_editor_hint()) {
					// Fil
					ProjectSettings::get_singleton()->set("rendering/environment/defaults/default_environment", "");
				} else {
					// Fil
					ERR_PRINT(RTR("Default Environment as specified in the project setting \"rendering/environment/defaults/default_environment\" could not be loaded."));
				}
			}