It's worthwhile quoting the summary given for the study you linked (this quote is not part of the study itself):

The basic idea is that nature, unlike a city, is filled with inherently interesting stimuli (like a sunset, or an unusual bird) that trigger our involuntary attention, but in a modest fashion. Because you can't help but stop and notice the reddish orange twilight sky - paying attention to the sunset doesn't take any extra work or cognitive control - our attentional circuits are able to refresh themselves. A walk in the woods is like a vacation for the prefrontal cortex.

Strolling in a city, however, forces the brain to constantly remain vigilant, as we avoid obstacles (moving cars), ignore irrelevant stimuli (that puppy in the window) and try not to get lost. The end result is that city walks are less restorative (at least for the prefrontal cortex) than strolls amid the serenity of nature.

In other words, it's not being in nature that's good for us. It's just not being in a city that's good for us :-)

So once we learn to build super-duper-optimized resorts for relaxation, we can put them in every other city block, and your reason for preserving real nature will become irrelevant.

there's the animal core of us that thinks "Oh, nature, I understand this."

Apart from all the cultural views you quote, my animal core panics in nature. It thinks: There's trees and bears and rocks and, and different looking trees! Where's my car? My cellphone? My backpack of food? My insulin pump supplies? HEEEELP!

Don't get me wrong, I love a walk in a quiet wood. But to feel relaxed I need a lot of safety and infrastructure. I'd guess quite a few modern city dwellers are more like me than they are like "oh, nature, I know this".