Adding Additional Fields to a Drupal Core Form (or any Drupal form) Using hook_form_alter

mymodule.module

<?php

/**
 * @file
 * mymodule module file.
 */

use Drupal\Core\Form\FormStateInterface;

/**
 * Implements hook_form_alter().
 */
function mymodule_form_alter(&$form, FormStateInterface $form_state, $form_id) {
  // Appends to the system - site information settings 
  if (str_contains('system_site_information_settings', $form_id)) {
    // Add an additional site_login_blurb fields and
    // save the values to system.site config.
    // Read current config values for site_login_blurb settings.
    $config = \Drupal::config('system.site');

    $form['site_information']['site_login_blurb'] = [
      '#type' => 'textarea',
      '#title' => t('Login Blurb'),
      '#default_value' => NULL !== $config->get('site_login_blurb') ? $config->get('site_login_blurb') : '',
      '#description' => t("Optionally, enter blurb text to be displayed above the site login form."),
    ];
    $form['site_information']['site_login_blurb_display'] = [
      '#type' => 'checkbox',
      '#title' => t('Display Blurb on Login Form'),
      '#default_value' => NULL !== $config->get('site_login_blurb_display') ? $config->get('site_login_blurb_display') : '',
      '#description' => t("If checked, display the login form blurb text."),
    ];

    // Append a submit handler to save custom blurb and checkbox value.
    $form['#submit'][] = '_mymodule_set_site_login_blurb';
  }
}

/**
 * Save the values to system.site config.
 */
function _mymodule_set_site_login_blurb(array &$form, FormStateInterface $form_state) {
  $config = \Drupal::getContainer()->get('config.factory')->getEditable('system.site');
  $config->set('site_login_blurb', $form_state->getValue('site_login_blurb'))
    ->set('site_login_blurb_display', $form_state->getValue('site_login_blurb_display'))
    ->save();
}