Yes, you have been taken by stereotypes. Autistic people absolutely have the capacity to empathize--We may feel and show empathy in different ways from others, or feel empathy for different reasons than others would expect. Many Autistic people are highly empathic, to the point of experiencing distress as a result.

https://medium.com/invisible-illness/autistic-people-are-not-emotionless-robots-91ae8df7c05c