I just tried installing the LINZ data importer plugin in QGIS 3.16.7 but got an error message (screenshot attached) and then figured out that I could successfully install it if I didn’t have a project already open, and then once installed, was only able to find the LINZ logo after restarting the whole program. Not sure if this is useful info to you or not. Possibly just issues you wouldn’t have any control over, with the version of QGIS I’m using? Simon suggested I email to let you know anyway.
