Psychopaths are not emotionless. Having diminished empathy is different from having no emotions whatsoever.
If this isn’t my absolute #1 RP/writing pet peeve, it’s pretty close.
It really goes for any mental condition, too. Schizophrenia =/= cartoony Hollywood multiple personalities (which ALSO isn’t how DID works, either, so that’s a double-whammy). Autism isn’t going to look like Rain Man. And so on.