Autistic people do have empathy, it just works differently. Most of them are actually really caring, just not very good at showing it. Nor at picking up others' feelings. But they do care about them in my experience.

Most of them I know will have more difficulty with this type of work, not less. Because they don't tend to process it as well. This includes myself as I do have some autistic tendencies. No way I could do this.

Autistic people often have stronger empathy that neurotypical people, sometimes much much stronger. Especially towards animals.