Actors are crazy, self-centered, overly emotional drama queens with potentially huge psychological issues.
I think there's a general assumption that being emotional and being intelligent are mutually exclusive. Either you're ruled by your heart, or your head. Since actors are associated with being emotional the same way, say lawyers are associated with being intellectual (and if we were having this same discussion about breaking down lawyer stereotypes we would be making the case for them not just being thinking heads. Seriously - in the musical "Legally Blonde" there's a lyric that goes: "Don't lawyers feel love too? Even if they do...") Actors are not unthinking wells of emotion. And we couldn't act if we were. We're in touch with our emotions, we're comfortable with them, but we're not ruled by them. We craft them. We're analytically emotional. Usually the first week at least of rehearsal is dedicated to what we call "text analysis" - literally intellectually analyzing the text and making conscious choices about what the character is feeling, and doing, and what we need to do to bring that across. It requires a great deal of empathy, yes, but also intelligence.