I'm skeptical (to put it a bit euphemistically) about TNR too. There is, though, evidence that at least on some time scales it is the least ineffective eradication method. Cats are territorial, and so if there's a cat in an area that prevents others from being in that area (and so prevents those others from having hunting grounds, and thus reduces their breeding potential). If a feral cat is simply removed from an area, those in other areas will be able to expand their hunting range and so have more offspring. Having a bunch of neutered feral cats around helps keep the non-neutered ones from reproducing.
I would be more supportive of TNR if it were one temporary tool in an overall program to get the number of feral cats down to zero. As a long time cat keeper (who recently captured two kittens in his woodshed*), I'd support some pretty serious regulation -- chipping and neutering of every captive cat, and criminal penalties for violations (in line with criminal penalties for other relevant wildlife violations such as killing migratory birds intentionally, which is currently a misdemeanor up to $15k fine and up to six months prison). Legit breeders could get licensed and have different regulations. Also, no cats outside off a leash or otherwise confined (as with a sturdy kennel or catio). Everyone who wants a cat should be allowed to keep one or more, and everyone who keeps one should be expected to keep it responsibly.
*We kept one, which is doing well though I've never had an eight week old kitten before and she's a fun little terror. The other was seriously injured -- two broken legs and a big wound, looks like a coyote might have gotten a hold of her -- but a tech at our vet agreed to take her in and nurse her back, and I have been told she's doing well.