While I'm not in a position to extrapolate on what life is like as a LBGTQ+ person in New Zealand, I did live here until I was 12 and moved back 7 years ago, so I can extrapolate on politics in general.
If I had to compare New Zealand to the United States in one sentence, I'd say that the main difference is that in terms of politics, the divide is economical rather than social. The more conservative major party (National) believes in tax cuts for the rich and tries to pinch pennies by chipping away at resources for schools and those on the benefit, while the more liberal party (Labour, currently the one in power) will do things like remove interest from student loans and offer a tuition fee free year for those going to university.
There are of course your social conservatives, but generally people don't seem to believe in pushing for and enforcing legislature that reflects their views. And once laws have been altered and they realise that, say, the same sex marriage bill didn't alter their day-to-day lives very much, they tend to just accept it. For example when Australia was having their marriage referendum, my grandmother commented on "what a fuss they're making, we have it (gay marriage) here and nobody died." While there are exceptions (such as Gloriavale) people usually aren't concerned too much with what's happening on the other side of the fence.
And therefore because it's an economic rather than a social divide, perhaps it's easier for people to find common ground. You do get your extreme conservative Christian groups, but generally it's believed that politics and religion should be kept separate, even if our national anthem does have lines like "God defend our free land" in it.
All that said, there are definitely issues here. We have one of the highest rates (if not the) of both infant mortality and youth suicide in the developed world, younger people are being increasingly priced out of owning property and it's estimated that 1 in 5 minors live below the poverty line. Racism is also abundant, especially against people of Asian descent and Maori/Pacifica. Islamophobia is also a thing. If anyone is seriously thinking of moving here, talk to me, maybe contact some New Zealand LBGTQ+ organisations, and I can hit up some gay colleagues for a better informed opinion.