Login form from shortcode doesn’t redirect after successful login

The problem you’re facing is because of the nature of the WordPress redirect. It uses what is called a safe redirect, namely with the function wp_safe_redirect(). This restricts redirects to local URLs only.

Your work around for this is the following where you jump into the login redirect filter and perform your desired redirection then, before WordPress does it for you.

Keep all your previous code as-is, and add the following:

add_filter( 'login_redirect', 'my_login_form_redirect', 0, 1 );

function my_login_form_redirect( $redirect ) {
    if ( !empty( $_POST[ 'redirect_to' ] ) && filter_var( $_POST[ 'redirect_to' ], FILTER_VALIDATE_URL ) ) {
        wp_redirect( $_POST[ 'redirect_to' ] );
    }
    return $redirect; 
}