RESOLVED:
Working with Microsoft support, I was able to determine what the issue was. The site in question is still using Classic SharePoint. The Forms for Excel only works on Modern SharePoint sites. More specifically it requires an M365 Group to be associated with the site.
This also explains why it Forms for Excel was present when testing on the SharePoint view of a Microsoft team. After all, creating a team automatically creates an M365 Group in the background to manage membership and permissions (and by extension the SP site is also associated with the same M365 Group).
It is possible to connect a classic SP site to an M365 Group - though this will also convert the site to a modern SP site (with a new default modern home page).
Here were two articles that explain the process in SP: