(hi Vincent.) I've lived in New Zealand for the last 12 years, and still also appreciate the the sounds of the birds every day. It never gets old, for me. The tui birds, bellbirds, saddlebacks, and grey warblers in particular.
When I lived in England for a year, I'd play recordings of New Zealand forests in the background because I was so homesick for these bird sounds.
The blackbirds and the tawny owls that live around my house remind me bit of the tui and the ruru sometime. But the tui birds are really unique and amazing! It's almost magical :) I remember listening to them one last time before hoping into the bus to the airport because I knew I would miss them a lot.
I also spent half a year on Steward Island during my first winter where there's a lot of kiwi birds and I still catch myself waiting for their calls in the forest next to my home in Europe at night haha!
When I lived in England for a year, I'd play recordings of New Zealand forests in the background because I was so homesick for these bird sounds.