Nor: More or Less

I've discussed the misuses of the word "nor."  I stress that the only correct use of "nor" is with "neither."  Well, guess what?  I'm not sure even that use is ideal.  Why?  Well, "neither" means "not either."  Let's substitute.  Take the sentence, "He is neither my brother nor my cousin."  Change it to, "He is not either my brother nor my cousin."  Oops--we've created a double (or 1 1/2) negative with both "not" and "nor."  How about, "He is not either my brother or my cousin."  Better.  Then why don't we say, "He is neither my brother or my cousin"?  Long-time custom, I guess.  I'd just as soon get rid of "nor" altogether.  Who's with me?