Well, the documentation does actually state that
CodeIgniter supports "flashdata", or session data that will only be available for the next server request, and are then automatically cleared.
as the very first thing, which obviusly means that you need to do a new server request. A redirect, a refresh, a link or some other mean to send the user to the next request.
Why use flashdata if you are using it in the same request, anyway? You'd might as well not use flashdata or use a regular session.