Core search module, change markup

I want to remove the Search results string added in the search page from line 119 of core/modules/search/src/Controller\SearchController.php.

if (count($results)) {
  $build['search_results_title'] = array(
    '#markup' => '<h2>' . $this->t('Search results') . '</h2>',
  );
}

I am able to change the search form above and the result list using preprocess_form function on the search form and preprocess_search_result on the search results.

Is there a preprocess function I missed, or can I use a custom template file?

You have to alter the route defined by the search module. In order to do so:

1. Define in your mymodule.services.yml file following:

    services:
      mymodule.route_subscriber:
      class: Drupal\mymodule\Routing\RouteSubscriber
      tags:
        - { name: event_subscriber }

2. Create a class that extends the RouteSubscriberBase class on /mymodule/src/Routing/RouteSubscriber.php as following:

<?php
    /**
     * @file
     * Contains \Drupal\mymodule\Routing\RouteSubscriber.
     */
    
    namespace Drupal\mymodule\Routing;
    
    use Drupal\Core\Routing\RouteSubscriberBase;
    use Symfony\Component\Routing\RouteCollection;
    
    /**
     * Listens to the dynamic route events.
     */
    class RouteSubscriber extends RouteSubscriberBase {
    
      /**
       * {@inheritdoc}
       */
      public function alterRoutes(RouteCollection $collection) {
        // Replace dynamically created "search.view_node_search" route's Controller
        // with our own.
        if ($route = $collection->get('search.view_node_search')) {
          $route->setDefault('_controller', '\Drupal\mymodule\Controller\MyModuleSearchController::view');
        }
      }
    }

3. Finally, the controller itself located on /mymodule/src/Controller/MyModuleSearchController.php

<?php
    namespace Drupal\mymodule\Controller;
    
    use Drupal\search\SearchPageInterface;
    use Symfony\Component\HttpFoundation\Request;
    use Drupal\search\Controller\SearchController;
    
    /**
     * Override the Route controller for search.
     */
    class MyModuleSearchController extends SearchController {
    
      /**
       * {@inheritdoc}
       */
      public function view(Request $request, SearchPageInterface $entity) {
        $build = parent::view($request, $entity);
        // Unset the Result title.
        if (isset($build['search_results_title'])) {
          unset($build['search_results_title']);
        }
    
        return $build;
      }
    
    }