Well, "said" isn't quite as invisible as "the" or "he" or "she" are, but it's a lot more invisible than the myriad alternatives that a lot of people try to use to avoid repetition.
Yet another reason following rules blindly is bad: there's this one "rule" that says that you shouldn't repeat the same word more than n times in a given page. (n varies depending on who you ask.) So you get stuff like Said Bookisms and the Burly Detective Syndrome, where people try desperately to avoid repeating "said" and the character's given name, respectively, despite the fact that those words are perfectly acceptible "repeaters" - it's long, unusual polysyllabics that you should avoid repeating too close together, just because they're a mouthful and hard to read. (The linked page lists "vertiginous" as an example of a word you don't want to repeat too often.)