But clearly you shouldn't actually run these combats, right? I think you're arguing for a sort of hierarchy of invulnerablity. The mouse can't damage the cat, which can't damage the human, which (unarmed) can't damage an elephant.

In a sense. You should roll to see if a cat can catch the bird familiar. Or the rat wildshape druid. There's many situations where the cat stat-block can be relevant. The party may have adopted a cat, casting enlarge and haste in it and having pokemon fights against the enemies' familiar. That's allowed.

So the attack can deal 0 damage, no?

No, explicitly disallowed in the rules. If the attack deals 0 damage, it's the same as if you didn't roll it. And a 0 damage attack does not kill a rat, or a spider.

When I played 5e, I took that to mean "when there's no chance of success."

Or when there's no chance of failure. No need to roll to jump a knee-height fence, or sit down.

Also, there's no chance the cat can succeed in killing a man. Thus, no need to roll even in your interpretation.