No, distilled water does not leach stainless steel. Stainless steel has a passivation layer of chromium oxide that makes it highly resistant to corrosion and possible reaction with water. Also, we have to understand that distilled water is not so salty as usual water. And as we know, the water of higher salinity is more corrosive.
Also, Joe Letorney pointed in his article that both samples of his tap water and the same water after the distillation system didn't have nickel. These samples were checked by the National Testing Labs of Cleveland, Ohio. It means that distilled water does not leach stainless steel.