So I just had a look at Interactive Brokers as many/most here using it, but at first glance I think that's not suitable for me: I'd first ask them whether NZ residents can be their clients, but neither on the main page nor on the main menu or the links at the bottom I could find a "contact" link, only info/links about their advantages re trading.

I googled "interactive brokers countries supported new zealand", and the answer is yes.
interactivebrokers.com - Open account country list

They are not hand holders. If that's what you want, look elsewhere. Most stuff is on line and automated.
But it's not exactly bad: their telephone support is generally gruff but extremely competent and knowledgeable, and their support tickets are generally answered very seriously.
I wouldn't give them up for the world. e.g., I do FX from time to time, and they plug you into the forex market directly. Current bid/ask on EURUSD is 1.08565 / 1.08570.
Not exactly the 3% spread you see at some other brokers!
I did a whole what of US equity trades today. Commissions were 0.00340% of the aggregate trade value.

The biggest downside:
Do not EVER get on the wrong side of their margin rules. They will [insert politically incorrect colourful metaphor] you.
The simple solution is not to use their margin, then everything is fine.

Jim