Finally got the NZIS webpage for the Guide to Studying in NZ to open up for me.

Under 'How do I apply' on page 4 it mentions that, if you study with certain education providers, renewing visas is made less difficult as you can apply on campus. I take it this is confirmation that you can get an original student visa, then renew it as necessary, presumably without having to go through the rigmarole of medical certificates or police checks again.