I have no problem with people wearing fur coats, just not made from dogs or cats.
It's quite a leap of logic to deny that the skinning takes place because Peta is against all fur coats.
That's not all they're arguing in the story, and they specifically do try to get people to avoid all fur from this one argument ("It is a disgusting industry and our advice is simply to avoid all types of fur."). They are making the specific charge that so many pelts from dogs and cats are making it into the commercial market as mis-labeled items that it's "unsafe" to buy any fur products for fear of buying something made from a dog or a cat. I'm pointing out that they have not made this case, and that their attempt to show their case with the "cat rug" is not only inadequate to their argument, it smacks of demagoguery and convinces me otherwise (that such widespread commercial penetration is indeed not taking place). Their example is something that specifically could never prove their argument, as it could convince no one of being anything other than what it is. Are cats and dogs being skinned somewhere in the world? Probably. Is it a problem in the commercial fur market, to the specific degree that one should avoid all fur, as they recommend? They have not shown this, and their (in my opinion) overreach tells me that it probably isn't the case. They simply have no credibility.