### 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)

_call_idle_callbacks();
#ifdef TOOLS_ENABLED
#ifndef _3D_DISABLED
	if (Engine::get_singleton()->is_editor_hint()) {
		
		String env_path = GLOBAL_GET(SNAME("rendering/environment/defaults/default_environment"));
		env_path = env_path.strip_edges(); 
		String cpath;
		Ref<Environment> fallback = get_root()->get_world_3d()->get_fallback_environment();
		if (fallback.is_valid()) {
			cpath = fallback->get_path();
		}
		if (cpath != env_path) {
			if (!env_path.is_empty()) {
				fallback = ResourceLoader::load(env_path);
				if (fallback.is_null()) {
					
					ProjectSettings::get_singleton()->set("rendering/environment/defaults/default_environment", "");
				}
			} else {
				fallback.unref();
			}
			get_root()->get_world_3d()->set_fallback_environment(fallback);
		}
	}
#endif 
#endif 
	return _quit;
}
void SceneTree::process_timers(double p_delta, bool p_physics_frame) {