It's not a huge mystery why this is an issue. Most land that axis roam on is private and either does not allow hunting or charges very inflated prices to hunt. The other factor is the Hawaii DLNR is terrible at herd management. I've been hunting axis during the archery season on Lanai for years but am contemplating not going back because of how unorganized the program is and because they absolutely failed in managing the 2020 drought that killed almost 2/3 of the herd. I'd love to hunt Molokai or Maui but unless you personally know a landowner who will allow you to hunt, be prepared to pay upwards of $5K just for one animal. That's crazy.
I'm sure those private landowners don't want to lose a huge revenue stream. I expect that nature conservation is going to take a back seat to that.
Up here in Upstate NY you don't even need to hunt. Sometimes I see a deer on the side of the road that has been freshly hit, these days I tell my wife and she tells our friends who will pick it up and cut it up for our freezer. Sometimes people beat us to get the rack but it feels so good to turn somebody else's $600 auto repair into $600 worth of meat.