New Zealand got a trade deal with the EU a few weeks ago that was worth a lot more to them than anything the Commonwealth could provide. The Commonwealth stopped being an important trading bloc decades ago.

I was in NZ when the UK joined the EU. It didn't have a lot of effect, links with the Pacific Rim had been getting more significant long before. As with the EU, freedom oF movement comes along with trade - NZ has more Chinese, Indian and Korean immigrants than British ones.