How to stop a user from updating the post date

As @t-f pointed out in his comment to question, you have an error on checking current user capability: 'delete_published_posts'. $post_author_id simply doesn’t exist.

After that, for your scope probably is a better hookin the filter wp_insert_post_data instead of the action save_post: because this one run when the post was already saved / updated, so if something is wrong you have to change and update again.

Hooking wp_insert_post_data filter, you can change the data that are being updated before update is ran, so this will improve performance.

add_filter('wp_insert_post_data', 'prevent_post_change', 20, 2);

function prevent_post_change( $data, $postarr ) {
  if ( ! isset($postarr['ID']) || ! $postarr['ID'] || current_user_can('delete_published_posts') )
    return $data;
  $old = get_post($postarr['ID']); // old post
  // prevent changing date
  $data['post_date'] =  $old->post_date 
  $data['post_date_gmt'] =  $old->post_date_gmt 
  // prevent sent to trash
  if ( $data['post_status'] == 'trash' ) $data['post_status'] = $old->post_status;
  return $data;
}