Moreover, it seems that insofar as you care about humans because they have certain psychological traits, you should care about any creature that has those traits. Since many animals have many of the traits that humans have, and some animals have those traits to a greater degree than some humans do, it seems you should care about at least some nonhuman animals.

Doesn't follow. If we imagine a personhood metric for animals evaluated over some reasonably large number of features, it might end up separating (most) humans from all nonhuman animals even if for each particular feature there exist some nonhuman animals that beat humans on it. There's no law of ethics saying that the parameter space has to be small.

It's not likely to be a clean separation, and there are almost certainly some exceptional specimens of H. sapiens that wouldn't stand up to such a metric, but -- although I can't speak for Qiaochu -- that's a bullet I'm willing to bite.