> For human consciousness to be "exceptional"... we would have to have some undefiniable "other" quality that could never be explained by science and is therefore necessarily supernatural?

No, it only means that science has limits on explaining the natural world. See Colin McGinn's argument for cognitive closure. He thinks we can't solve certain problems in philosophy because we lack the correct cognitive machinery. Consciousness being a primary example since it proposes a subjective/objective split where science only explains the objective. Nagel argued along similar lines in his paper, "What Is It Like to Be a Bat?".

There's no need to invoke the supernatural. It just means some natural things may be beyond our capability to understand. Why would we have evolved to understand everything?