Admin Toolbar Extra Tools items on /admin/index

Here's an easier way to do this:

1). Copy admin-block.html.twig from core/modules/system/templates/admin-block.html.twig into your admin theme.
2). Add this twig:

{% set class = block.title %}
{% set class = class|replace({' ':''}) %}
{% set class = class|lower %}
{%
  set classes = [
    'panel',
     class,
  ]
%}

Then, the <code><div{{ attributes.addClass(classes) }}>

will include the class name of the region, in this case, admintoolbarsearch

Then, in your admin theme, just

div.admintoolbarsearch {
display:none;
}

The list that admin_toolbar_tools generate on admin/index is really cluttering the UI and the link titles are often confusing.

If you want to stop the output of admin_toolbar_tools before this actually is put on the site, you could use a Route Subscriber and override the core controller to exclude the links from admin_toolbar_tools.

Like so:

your_module.services.yml

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

src/Routing/RouteSubscriber.php

<?php

namespace Drupal\your_module\Routing;

use Drupal\Core\Routing\RouteSubscriberBase;
use Symfony\Component\Routing\RouteCollection;

class RouteSubscriber extends RouteSubscriberBase {

  protected function alterRoutes(RouteCollection $collection) {
    if ($route_index = $collection->get('system.admin_index')) {
      $route_index->setDefault('_controller', '\Drupal\your_module\Controller\AdminController::index');
    }
  }

}

src/Controller/AdminController

<?php

namespace Drupal\your_module\Controller;

/*
 * hide the admin toolbar extra tools entries on admin/index as it clutters the UI.
 */
class AdminController extends \Drupal\system\Controller\AdminController {

  public function index() {
    $output = parent::index();
    if (isset($output['#menu_items']['Admin Toolbar Extra Tools'])) {
      unset($output['#menu_items']['Admin Toolbar Extra Tools']);
    }
    return $output;
  }
}