I set wp_set_auth_cookie but wp_validate_auth_cookie returns false

Question

If you want the login works without having to reload the page (or redirect to somewhere on the site), then you would want to immediately set both the auth and logged-in cookies, i.e. set them in the superglobal $_COOKIE array. And you can do that via the set_auth_cookie and set_logged_in_cookie hooks:

add_action( 'set_auth_cookie', function ( $cookie ) {
    $cookie_name = is_ssl() ? SECURE_AUTH_COOKIE : AUTH_COOKIE;
    $_COOKIE[ $cookie_name ] = $cookie;
} );

add_action( 'set_logged_in_cookie', function ( $cookie ) {
    $_COOKIE[ LOGGED_IN_COOKIE ] = $cookie;
} );