Hi Elliot,
I’d just like to chime in and say that have_rows doesn’t seem to generate any output on the front-end when used within an Options page (the field data is stored just fine in the database though).
I’ve tried your api-template.php hotfix, but sadly, that doesn’t seem to have done the trick 🙁
For the sake of reference, I am running WP 4.3.1 and ACF Pro latest.