Since 2016 when a more bowler-friendly era started to develop, Tests in New Zealand have recorded the highest runs-per-wicket on the circuit other than the notoriously batsman-friendly Pakistan/UAE. The 2011-2015 NZ summers were not overly different.
New Zealand were the only team that didn't see their batting averages drop in the peak bowling era period 2017-2021 (some of that is also down to opposition schedule) and they all (Latham, Kane, Taylor, Nicholls, CDG etc) had major home/away disparities; so it is likely New Zealand has been a good to very good place to bat through Kane's career. In terms of eye test, I watch a lot of Tests in NZ and yeah, it is generally very hard work for the bowlers after the new ball period, especially after day 1.
Kane does bat top 3 though which is tougher; often has to negotiate early swing. His home record is very good and deserves credit.
As for Cook's record there, he wasn't the best player of lateral movement. But if he played half his tests in New Zealand like Kane, I suspect he would be fairly successful. Facing ill-equipped attacks on a frequent basis that didn't grow up in those conditions. Latham still averages above 40 there and Cook is a vastly better player than him. Thus, Cook not doing great in NZ doesn't necessarily mean it hasn't been a good place for a batsman to have their home tests in, nor does it mean he would fail there in Kane's matches.