Throwing in my lot for .308.
I shoot a Kimber MA in .308 22". While I can't say with certainty that it would suit most NZ scenarios, it's performed admirably in every NA scenario I've used it in-- elk in the Rockies, Aoudad hunting in West Texas canyons, to pissing rain in 1º temps in Alaska. As a cartridge, it's hard to beat in terms of availability, cost, and variety of loads. Mileage may very, but my rifle shoots any virtually any 150 gr. factory load of reasonable quality with enough accuracy and consistency for me to be reasonably confident taking a shots out to 300m. If I'm hunting larger/tougher game (as opposed to Texas whitetail), or know I'll be taking longer shots, Nosler Accubond 165gr has been my go to (but not exactly the most economical round to shoot on a regular basis). The gun was finicky with 180gr initially, but performs fine with the heavier stuff after proper break-in and glass bedding.
With scope, rings, 1.5" Arca rail, and factory brake, the entire rig runs about 3.18kg. Recoil is absolutely manageable, as is maintaining sight picture with the right technique. More so when shooting suppressed. I'm almost always shooting supported, either from a bag with lightweight fill on top of my pack, or from a Sirui tripod with an Arca plate if the scenario allows. I've found that maintaining a few pounds of pressure on top of the scope works well for me. I'm also using a bag, binos, or whatever I have available to support my elbow or the butt, which mitigates some of the pitch from recoil. An under appreciated aspect of maintaining sight picture is, IMHO, exit pupil size and field of view. I'm running a 2.5-15x42 scope (more mag than needed most of the time), and it's certainly easier to keep your eye on the target at less than full mag.
I think the 6.5 Creedmoor absolutely could fit your criteria as well, and with the right load would be adequate for most big game applications. It certainly beats out .308 in terms of recoil. That said, owning lightweight rifles chambered in both, I'm more often reaching for the .308 simply for the additional downrange energy it offers.
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