In New Zealand, the local EFTPOS network is basically a home-grown network that the major NZ banks established in the 1980s's and have continued operating on a shoe-string budget ever since. For the most part merchants only have to pay a low fixed cost to access the network, whereas customers usually don't pay a fee (as most banks either bundle free EFTPOS transactions in a fixed monthly fee package or even offer it for free entirely for certain account products).
Visa/Mastercard on the other hand has Merchants paying 1-3% of all transactions as well as customers paying an annual fee (usually around $25 or so for standard non-reward cards but can go up to hundreds of dollars for higher end cards) for many card products (so a tidy profit from the merchants as well as some annual top-up from the customers).
From a typical New Zealand merchant's perspective it doesn't always make sense to offer Visa/Mastercard if they can decide to only accept EFTPOS cards (which almost everyone in New Zealand has) and keep the 1-3% that they would otherwise lose to Visa/Mastercard.
Visa/Mastercard offer more modern benefits (e.g. Paywave) which is not offered by EFTPOS which I suspect is partly due to the fact New Zealand's EFTPOS network does not generate much of its own income (just the small fees merchants pay to access the network I believe) so they just continue with the minimum amount of investment.
Personally I have an zero-fee American Express card that earns Air New Zealand Airpoints. As a fallback at merchants that don't accept Amex, I have a Wise (ex TransferWise) Mastercard on my phone (for NFC payments) as well as a physical Kiwibank Visa card. When none of these are accepted I fall back on to my EFTPOS card which is more or less on par with cash in New Zealand. I estimate around 50% of my expenses are on Amex, 10% on Visa/Mastercard and the balance through EFTPOS with the occasional cash for markets and the like.