Hi Troy,
Those search elements are using wordpress search function.
You can use wordpress filters to exclude your private pages.
You can try adding the code below in your child theme’s functions.php file.
//Exclude pages from WordPress Search
if (!is_admin()) {
function wpb_search_filter($query) {
if ($query->is_search) {
$query->set('post_status', 'publish');
}
return $query;
}
add_filter('pre_get_posts','wpb_search_filter');
}
Please add it after this line
// Additional Functions
// =============================================================================
For more information, you refer to the links below
https://codex.wordpress.org/Plugin_API/Filter_Reference
Hope that helps