This is why I wrote 'PETA-like'.
And falconry being 'nature' is debatable.
As a falconer, the best description of falconry I've come across is "hardcore birdwatching". Not that I'd "define" it as such, but it really is the best "description" I've heard.
As for it being "nature"? Maybe, maybe not. Personally, I consider the human race to be a part of the ecology of the planet. As such, I see people (and what they do) as a legitimate part of "nature".
Now, would a raptor pair up with a human? Probably not as a hunting "partner"...
However, they do "use" people (and predators, and other large animals and herds of animals) and their propensity to (often accidentally) flush quarry out into the open. It's not uncommon for raptors to take advantage of that while hunting. Flying a falcon in "waiting on" style and flushing ducks, grouse, or pheasants under it is about as close to a mimicry of this event as it gets.
Having watched wild redtails hunting squirrels, mice, snakes, pheasants, etc. (while I was unobserved), and having hunted literally hundreds of squirrels with my "wild but trained" redtails I can attest to the fact that there is surprisingly little difference between the "natural" event and falconry. I could elaborate and give other examples of hunting with falcons and accipiters as well, but it would be redundant for the most part...
It's also not uncommon for raptors to "use" human alterations of the environment either to hunt gathered quarry (such as the bird feeder or altered habitat drawn sparrows, etc., that cpxxx saw captured) or to ambush quarry (such as a redtail sitting on a telephone pole.
Now, is it "natural" for a redtail to hunt from a telephone pole? Or to hunt mice under that pole that are more-easily accessible due to the placement of that pole and the mowing habits of humans?
What if the redtail is hunting from a tree that was planted by humans? Or from a "naturally-occurring" tree that is only an acceptable perch because it borders grasslands that it wouldn't naturally border due to the human division of massive tracts of "redtail-unfriendly" forest into smaller "redtail-ideal" woodlots?
Agricultural practices have opened massive amounts of previously unsuitable habitat for redtails, and provided mile upon mile of ideal hunting habitat for assorted prey (in the form of wide grassy highway borders, complete with perches, and even fences and fast-moving traffic to ensure that they aren't disturbed by people). Is a redtail hunting along an undisturbed section of highway an example of "nature" in action?
Is it "natural" for humans to use/exploit/benefit from animals?
Yup... It's all debatable.