You are conflating the idea of a stereotype, and the idea of a generalization. A stereotype is a particular type of generalization that infers one set of personal traits, such as intelligence, personality, or interests, from another, generally unrelated set of traits, such as gender, ethnicity or sexuality.
The usual argument against stereotypes is that the kinds of connections they make are typically spurious. Even if you believe that, however, that does not mean that all generalizations are therefore invalid. There are many many types of generalizations that have nothing to do with stereotypes.
Your friend's argument is actually a bit more nuanced --she isn't arguing that stereotypes are wrong, but that they are socially harmful. As it turns out, this is actually a topic with a fair amount of scientific research behind it, which indicates that stereotypes do indeed consistently and significantly impact us, in the statistical aggregate, even if we do not consciously affirm them. So your friend's argument is underwritten by science, while yours is not.