That people need more than food and drink, they also need entertainment? And working for that is no less valuable than farming? But how is that related to cats?

I guess cats can provide companionship, which is valuable to a nomadic society. But so can dogs, and dogs have a bunch of other benefits. Cats are more useful when mice and rats start eating your food stores, and happily that's also when humans become useful to cats

Yes companionship.

But your answer with mice and rats add clear benefit to it too.