Like @odulhin mentioned, you might have forgotten the CSS snippet.
Alternatively, just on the off chance you or someone else runs into the same niche issue as I did: if you “Add a new note from annotations” in Zotero, the imported annotations won’t show up in call-out boxes at all. They’ll just look like plain text. The Zotero Integration and this template draw from the annotations themselves, so if you’ve got an Annotations note attached to the ref you’re trying to import, you can probably delete it.
If you do that and then find that no annotations are imported at all, check how many PDFs you have attached to the reference in Zotero. As far as I can tell this template only draws annotations from the top one, so delete any duplicate PDFs until your annotated PDF is the only one remaining. Those two steps fixed it for me.
I would be interested in seeing a template that could pull annotations from multiple PDFs, since having duplicates appears to be a relatively common problem for me.