add_rewrite_rule doesn't work when trying to add a new parameter at the end of URL

Question

I wish to add a userq parameter at the end of a URL with the username such as john. I wish to change,

example.com/user/john

to,

example.com/user/index.php?userq=john

I tried this and this and they don't work.

add_action('init', 'user_profile_rewrite');
function user_profile_rewrite() {
        global $wp, $wp_rewrite;
        $wp->add_query_var('userq');
        $rule = '^user/([^/]*)/?';
        $rewrite = 'user/index.php?userq=$matches[1]';
        add_rewrite_rule( $rule, $rewrite, 'top' );
        $wp_rewrite->flush_rules();
}

I think the error has something to do with user/index.php? so I tried removing the word user by changing to index.php?... but the URL remains like the original. How to solve this?