I think that the Aztec are the weakest civilization. Personally, I don't find the extra gold that much helpful.
I think you're wrong. I think they're ridiculously powerful at the start, even though I haven't used them yet. I played a Free For All private match with 2 friends online on my PS3 the other day, and one of them was using the Aztecs. I was unfortunate to start out in a poor spot and very near his starting position. On Turn 1, he used his starting gold to rush build his first Warrior unit and start exploring. He found me and claimed my capital before my first Warrior unit was built. Game over.
On other matches (he uses aztecs almost exclusively on MP), he's able to build his Civ much faster than everyone else. Turn 1 he rush builds Warriors and is able to start exploring and I think he has enough gold left over to rush build another Unit by Turn 3 or 4. Using that tactic, he has 2 Warrior units built that can be exploring the area around him, claiming land naming rights and friendly/barbarian villages before other Civs even have their first Warrior unit up. Because of this, even though he spends much of his starting gold, he still gets his bonus (100 gold) settlers way before anyone else, and if you were unlucky enough to start near him, by the time you start exploring you find there is nothing in your area left to claim. So, you lose out on early exploration gold and it takes until much later in the game to get your bonus settlers (which is HUGE), and instead of building enough Warriors to defend your turf and explore early, or a Galley to get to island exploration, you end up spending valuable time early on building a settler unit, which slows everything down. Science production and starting city growth (hurts less if you're using Romans, which I hardly ever do now that I got the full game) are stunted so badly, if he finds you early in the game and is aggresive enough, your game is essentially over.
So, I would strongly disagree that the Aztecs are the worst Civ, by far.