Injecting service into another service does not work

Question

I have to override pagecache service in order to have some custom validation logic to be executed for anonymous users. The issue is that I have the domain module's service domain.negotiator to be injected into pagecache service, but when I call the domain.negotiator service it is always null no matter what I do. I think it has to do with something with priority. I can confirm the overridden service is being executed.

Code to override PageCache service

/**
   * {@inheritdoc}
   */
  public function alter(ContainerBuilder $container) {
    // Overrides language_manager class to test domain language negotiation.
    $definition = $container->getDefinition('http_middleware.page_cache');
    $definition->setClass('Drupal\hcl_login\StackMiddleware\StaticCache')->addArgument(
      new Reference('domain.negotiator'),
    );
  }

Overridden service

/**
   * {@inheritdoc}
   */
  public function handle(Request $request, $type = self::MASTER_REQUEST, $catch = TRUE): Response {
    $domain = \Drupal::service('domain.negotiator');
    $x = $this->domainNegotiator->getActiveDomain();
    $currentDomain = \Drupal::service('entity_type.manager')->getStorage('domain')->createHostname();
    $response = parent::handle($request, $type, $catch);
    return $response;
  }