Dewormers are not antibiotic's, they do not kill any bacteria, good or bad, they kill parasitic worms. As far as using something labeled for another animal? I do not know of any species specific dewormer. Many wormers are used for many different animals, including humans unlucky enough to pick up some parasite, the active ingredient is the same. The few that are labeled for chickens are woefully inadequate, they only treat roundworms and there are many parasites chickens can get other then roundworms. That is why most people use other wormers "off label". Also, not all parasites will show up in the droppings, not as the worm itself or the eggs. So a clear fecal sample does not mean the birds don't have worms. As far as letting nature take it's course? Unfortunately we don't live in a perfect world of green pastures and rainbows, nature is often incredibly cruel and wild creatures often live a very short life and die a hard death compared to our pampered pets.
Yes, chickens and any animal can handle a certain amount of parasites. But how do you know when the numbers are out of control? Do you wait until the birds start showing poor condition, loosing weight, having diarrhea etc? Often times when they've gotten to that point it's hard to bring them back, parasites do a lot of internal damage.
As far as DE as a wormer? Tried that years ago when it was all the rage, chickens got worms. So I deworm my birds a couple times a year, parasite loads are kept under control very easily. But it all boils down to a personal choice, what you want for you and your birds. Research poultry parasites thoroughly, don't rely simply on what you read on a forum, we all have our own opinions. But do your own research and base your choice on that.