fix: [Stateful: Indices: Overview page] Missing focus on New button a…

…fter closing or cancelling Generate API key dialog (#196905)

Closes: #196174

## Description 

Focus should be visible on any accessible element and in order when
navigating through the elements. Especially for the users using
assistive technology, only keyboard.
For the dialogs, after closing them, focus should be visible on the
element which opened it.

## What was changed: 
1. Cleaned up `generate_api_key_panel.tsx`. No need to open
`GenerateApiKeyModal` from this level. Same logic duplicated in
`APIGettingStarted` inner component

## Screen


https://github.com/user-attachments/assets/46a23fdd-ba54-4798-9cb7-a78e28d447b8

x-pack/plugins/enterprise_search/public/applications/enterprise_search_content/components/search_index/generate_api_key_panel.tsx

@@ -7,27 +7,20 @@
 
 import React from 'react';
 
-import { useActions, useValues } from 'kea';
+import { useValues } from 'kea';
 
 import { EuiEmptyPrompt, EuiFlexGroup, EuiFlexItem, EuiPanel } from '@elastic/eui';
 
 import { i18n } from '@kbn/i18n';
 
-import { GenerateApiKeyModal } from './components/generate_api_key_modal/modal';
-
 import { APIGettingStarted } from './components/getting_started/getting_started';
 import { IndexViewLogic } from './index_view_logic';
-import { OverviewLogic } from './overview.logic';
 
 export const GenerateApiKeyPanel: React.FC = () => {
-  const { isGenerateModalOpen } = useValues(OverviewLogic);
-  const { indexName, isHiddenIndex } = useValues(IndexViewLogic);
-  const { closeGenerateModal } = useActions(OverviewLogic);
+  const { isHiddenIndex } = useValues(IndexViewLogic);
+
   return (
     <>
-      {isGenerateModalOpen && (
-        <GenerateApiKeyModal indexName={indexName} onClose={closeGenerateModal} />
-      )}
       <EuiFlexGroup>
         <EuiFlexItem>
           <EuiPanel hasBorder paddingSize="xl">