Not american, so that link doesn't apply to me. Here everyone is a potential donor, and the family decides. My immediate family knows full well that my organs are FFA when I don't need them anymore.
In the US, it's actually up to the next of kin, too. When you die, they technically own your corpse. Registering basically just makes a record of your opinion. Even if you aren't registered, I think your family can decide to donate the spare parts