[Background search] Remove unused code (#240405)

## Summary

Closes #240226

- Removes connected_search_session_indicator as it has been replaced by
the flyout + send to background button.
- Removes the documentation link as it's no longer used.
- Replaces the EuiIconBackgroundTask from the background-search package
for the backgroundTask icon from EUI

### Checklist

Check the PR satisfies following conditions. 

Reviewers should verify this PR satisfies this list as well.

- [x] The PR description includes the appropriate Release Notes section,
and the correct `release_note:*` label is applied per the
[guidelines](https://www.elastic.co/guide/en/kibana/master/contributing.html#kibana-release-notes-process)
- [x] Review the [backport
guidelines](https://docs.google.com/document/d/1VyN5k91e5OVumlc0Gb9RPa3h1ewuPE705nRtioPiTvY/edit?usp=sharing)
and apply applicable `backport:*` labels.

---------

Co-authored-by: kibanamachine <[email protected]>

Author: AlexGPlay

src/platform/packages/shared/kbn-background-search/index.ts

@@ -8,4 +8,3 @@
  */
 
 export { BackgroundSearchRestoredCallout } from './src/components';
-export { EuiIconBackgroundTask } from './src/components';

src/platform/packages/shared/kbn-background-search/src/components/eui_icon_background_task.tsx

@@ -8,4 +8,3 @@
  */
 
 export { BackgroundSearchRestoredCallout } from './background_search_restored_callout';
-export { EuiIconBackgroundTask } from './eui_icon_background_task';

src/platform/packages/shared/kbn-background-search/src/components/index.ts

@@ -15,7 +15,6 @@ import type { TopNavMenuData } from '@kbn/navigation-plugin/public';
 import { useBatchedPublishingSubjects } from '@kbn/presentation-publishing';
 
 import useObservable from 'react-use/lib/useObservable';
-import { EuiIconBackgroundTask } from '@kbn/background-search';
 import { UI_SETTINGS } from '../../../common/constants';
 import { useDashboardApi } from '../../dashboard_api/use_dashboard_api';
 import { confirmDiscardUnsavedChanges } from '../../dashboard_listing/confirm_overlays';
@@ -195,8 +194,7 @@ export const useDashboardMenuItems = ({
       backgroundSearch: {
         ...topNavStrings.backgroundSearch,
         id: 'backgroundSearch',
-        // TODO: Replace when the backgroundTask icon is available in EUI
-        iconType: EuiIconBackgroundTask,
+        iconType: 'backgroundTask',
         iconOnly: true,
         testId: 'openBackgroundSearchFlyoutButton',
         run: () =>

src/platform/plugins/shared/dashboard/public/dashboard_app/top_nav/use_dashboard_menu_items.tsx

@@ -91,7 +91,6 @@
     "@kbn/controls-schemas",
     "@kbn/controls-constants",
     "@kbn/presentation-util",
-    "@kbn/background-search",
     "@kbn/react-query"
   ],
   "exclude": ["target/**/*"]