pre_get_posts Tax Query not working for custom author page

Don’t run a new query in the template, modify the main query before it’s run via the pre_get_posts action in the theme’s functions.php file.

function wpd_author_query( $query ) {
    if ( $query->is_author()
        && $query->is_main_query() ) {
            // your code to set $current_user_name here
            $query->set( 'meta_key', '_writer_relation_added_date_' . $current_user_name );
            $query->set( 'orderby', 'meta_value_num' );
            $tax_query = array(  
                array(
                    'taxonomy' => 'writer',
                    'field' => 'name',
                    'terms' => $current_user_name
                )
            )
            $query->set( 'tax_query', $tax_query );

            // EDIT
            // unset the requested author
            unset( $query->query_vars['author_name'] );
    }
}
add_action( 'pre_get_posts', 'wpd_author_query' );

You can then run the vanilla loop in the default author template, and save an extra query in the process.