I've got a theory about why small wildcats domesticated: touch. Small cats love to be touched. They cannot get it from other cats. But they can from people.
The early theory for domestication was the mutuality of keeping grain stores free of vermin. This was based on the paintings found in the remains of ancient Egypt. Nice idea. New evidence shows domestication happened over 100,000 years ago—when grain stores, if they existed, were negligible.
We think that lap cats are a modern phenomenon. Maybe not. Maybe it was practices like Puritanism that kept hands off cat bodies. People have a long track record of self denial. That could explain why people denied themselves the deleriously tactile pleasure of stroking a cat.
Perhaps early people weren't so inhibited.
Humans enjoy tactile pleasures. We like fur pelts and velvet fabrics for the way they feel when touched. Certainly paleolithic peoples enjoyed this. Why wouldn't they have embraced the small cat, a creature the size of a human baby with a flexible body, an animal easily held and cuddled, as a member of their extended family? An animal that could take care of itself and its young while returning "home" regularly for laps and pettings—just like today's house cat.
Cats also offer affection, comfort, and companionship that may have been as attractive to people 100,000 years ago as it is to me today. Cats have a way of reinforcing the essential humanity of people. Surely they've been this way for a long time.