I got the feeling that he likes his dead dog more than the wife he met on... Tinder. If my wife had diabetes, or really any medical condition, I would never reveal that in and article written in the New York Times. Her nagging him and having diabetes appear to be her only qualities from his description in the article.
All I get from this is that Dusty was an awesome celebrity dog who died and this guy's wife whom he met on Tinder sucks and is diabetic. The structure also implies that he put his dog down so that he wouldn't get in between him and his bullying wife.
I would never write anything disparaging or potentially embarrassing about my wife in a public article.