Shortcode for registration form not working for new users

Actually, that’s the correct behavior. Until a user is logged in, neither WordPress nor s2Member has any way of identifying who the user is, so there is no way that any profile can be shown until the user is logged in.

I was under the impression that users can *register* using the same form/shortcode (on a custom page)?

No, here’s the KB article: https://s2member.com/kb-article/s2member-profile-shortcode/

The default means of registering uses WP’s own registration page. Paid registrations use buttons or forms designed for the purpose: https://s2member.com/kb-article/what-is-the-difference-between-the-membership-options-page-and-the-register-page/