Well, there is "Ghost and the Darkness" with lions, "The Edge" with a bear, and a Korean movie called "The Tiger" that I haven't watched but saw a preview to recently. They are classified as action/drama, drama, and adventure/drama respectively. Is a movie where a tiger stalks, kills, and eats deer horror? Or does it only become horror when a tiger stalks, kills, and eats people? Because there are lots of documented cases of very prolific man eating Bengal tigers in India. These cases usually arise from an injured or aging tiger. People just happen to be easy to stalk and kill. And in India, there is no shortage of people. These aren't cases of horror or psychological torment, it's just nature. Wrong place wrong time, like a car crash.
I guess the point I'm trying to make is this. A movie where a person stalks and kills other people can very easily fall into the horror genre depending on how it's portrayed. People are self aware, animals are not. A portrayal of a tiger or the alien from the Alien movies killing people is different. They don't understand they are frightening people or causing torment. They are not self aware, they are just trying eat and/or reproduce. They are trying to fulfill life processes. Michael Meyers, Jason Vorhees, and Freddy Krueger are not trying to fulfill life processes, neither is Buffalo Bill, or the people that pay to kill and torture other humans in the Hostel movies.
Also, Alien doesn't know that humans might read into its reproductive cycle as reflecting some aspect of sexual rape anymore than a botfly or a tapeworm knows humans might link their intrusion into us metaphorically in some way. It's just how they live and reproduce. Although I admit some of those botfly larvae videos on YouTube are pretty horrific. But not in a psychologically tormenting or blood curdling way. In a "damn nature, you scary" kind of way.
But I do think you make one very important point. If there is a horror aspect to Alien, it lies in the plot detail in which "the company" has ordered Ash to return with the Alien and has deemed the crew expendable. The idea that certain people offered up other people to be sacrificed so that some creature could fulfill its life cycle through them, thus giving the other people access to specimens of that creature would definitely classify as physical and psychological horror and torment. And as you say, setting this up actually encompasses a lot of the movie. But at the end of the day Alien is no different than if a giant botfly was set loose on a spaceship. That's why I think it is Sci-fi, because it basically deals with "what else is out there, what form might it take, and how will we interact with it" as opposed to what does it mean to suffer, or be terrorized, or tormented.