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register_post_type() has a argument exclude_from_search that you can set to true for your to-be-exempt post types, so you don't have to dabble with a "hidden" term work around.

Alternatively you can hook into the search query itself via pre_get_posts and modify which post types you're after:

add_filter('pre_get_posts',function ($query) {
    if ($query->is_search && !is_admin() )
        $query->set('post_type',array('post','page'));
    return $query;
});

If you were set on the taxonomy hiding, you'd do somthing like:

add_action( 'pre_get_posts', function ( $query ) {
    global $wp_the_query;
    if( $query === $wp_the_query && $query->is_search() ) {
        $tax_query = array(
            array(
                'taxonomy' => 'category',
                'field' => 'slug',
                'terms' => 'hidden',
                'operator' => 'NOT IN',
            )
        );
        $query->set( 'tax_query', $tax_query );
    }
});