That's another problem with it
Animals are only inferior in intelligence. They still have feelings and everything
Intelligence is directly related to feelings, which in their most complex forms require, first, a level of neocortical development only to be found in humans; and, second, specific cultural contexts that shape the feeling and how people experience it (a cultural complexity missing in other animals). Non-human animals experience from very basic emotions (even if it's only surprise) to feelings in the most intellectually developed (e.g. other hominids), but certain sentiments are reserved to humans as the most intelligent beings.
"Proper" is a keyword here. Unlike "sociopathy", which was deemed obsolete, "psychopathy" is still used in studies, forensic and legal context.
That's because there is an argument on what it exactly is, a separate disorder, extreme/more violent version of ASPD or as you said, that it doesn't actually exist but is just combination of Cluster B symptoms.
But a lot of personality disorders share similar symptoms, so that doesn't exclude it immediately
It gets to a point that it feels irrelevant to talk about "psychopaths", which is what I'm leading to. Even what you'd call the most basic assumptions about such profiles, like their lack of empathy, are heavily doubted or discussed (the empathy issue I mentioned was, indeed, one of the examples given during my training as psychologist). Those studies, forensic and legal context you mention will usually stick to the vagueness of "psychopathic traits" or similar, many times on a weak basis (e.g. many papers I've come across these years that talk about psychopathy use tools lacking enough reliability, such as the Dark Triad Test which, as far as I recall, showed strong overlap between its constructs; or simply choose "psychopathic" in absence of a better diagnosis). When the psychopathic profile can be better explained by using recognized labels (for example, I see no necessity in separating psychopathy from ASPD on the basis of it being a "extreme/more violent" version of it instead of simply including it as part of its spectrum), "psychopath" as a diagnostic becomes, at best unnecessary, at worst a buzzword abused by the average people that comes in handy to pathologize whatever they find unmoral.