Hi, I have a New Zealand too so I know how high they can jump!
I think his hormones are in full force so neutering him would be the best to cut down on them. It won’t completely get rid of them but it can cut them down up to 60% but of course with age the also settle down so that is part of it.
If you can put him in a pen when you are not home with the top covered with fleece of something, you can zip tie it, then that will give him plenty of space to still roam around compared to the dog crate

I definitely think neutering will help to calm him down, but it’s obviously difficult to say to what extent. Do also remember there’s often a ‘hormonal surge’ after neuters, so things seem to get temporarily worse before they get better.

I don’t think you need to feel bad about caging him for short periods, for his own safety. It sounds like he gets a great amount of time out, but if he is causing such destruction, he’s also a danger to himself, so it might be best to keep him in a large cage and pen or single room when you’re not around to monitor him. Is there a room (such as the kitchen for example?) that is more bunny proofed than others? It might be good to get him a home base, and a room that you shut him in when you’re not at home/ or are in bed.

Just to give a point of reference, Clover, my 18 pound, American Chinchilla, can make a 38″ vertical leap with no effort at all. The only reason she hasn’t jumped higher is I don’t have anything higher for her to land on. Dawn, our Dwarf, makes easy 30″ jumps onto the couch daily. I think rabbits can jump a LOT higher than people think, but they won’t try to jump anywhere they can’t see to land on.