The advantage is light weight in the gun. My buddy is pretty hard core and gets a long way into difficult terrain. A 3 1/2 pound gun is much, much easier to pack into where he's typically going than a more typical 6 1/2 to 7 1/2 pound shotgun. As I mentioned in my first post, he likes the folding Yildiz. They pack up extra small. They are also relatively cheap, compared to his (many) other shotguns and he is happy to experiment with custom camo coatings on the whole gun. He thinks long range shots from any gauge are dumb and makes sure the birds are well within 40 yards before squeezing the trigger. It's not about bragging rights......it's about a guy with 35 years experience turkey hunting developing the best tool for the job, as he sees it.
Also, re your point about TSS being very expensive. It sure would be, comparatively, if one were using it at the range, waterfowl hunting or wingshooting on a regular basis. But given the number of shots taken for each turkey, typically one or two, worrying about the price of TSS is a false economy, given it's performance advantages, IMHO.
I also know people who regularly hunt wild quail with a .410. Actually much harder to do than shoot a turkey with a .410. You just have to be good. I'm not so don't try.