And I think that's a shame, as the AI won't be nearly as focused at early attacks as a human player, so the huns will be used only by people who play Civ like a deathmatch, or badly as an AI.
It is a shame, because the Huns do play very badly as an AI - it's not helped that the AI is hampered in its early rush (and city defence) by its conviction that Battering Rams are a type of Spearman, and so their ideal uses include protecting the Great General, preferentially moving into rough terrain to maximise their defensive bonus, and standing their ground against enemy melee units or as a screen to protect the Horse Archers.
In games where I've faced them Attila rarely lasts long, and when he does he has difficulty expanding as his AI is apparently programmed to expect its early cities to be captured capitals (since it razes everything else anyway) and doesn't seem able to adapt to a strategy that actually involves settlers (the few times I've encountered Hunnic settlers they tend to loiter around at the edge of his territory unsure of their purpose, until a kind soul visits and directs them to work on my farms and in my mines).
And what makes it even more of a shame is that I actually like having Attila in games - his AI has attitude, if he does capture cities the whole razing thing is an interesting dynamic since other AIs never do (except CSes in vanilla, but in G&K my experience of CSes is that they're so passive they're very unlikely to even try to take cities any more), and his schoolyard bully routine is spot-on - I'm always getting messages that CS X has been bullied by the Huns, sometimes accompanied by Attila popping up saying "Oh, I'm sorry, I seem to have inadvertently bullied a city-state under your protection, please forgive me. And don't mind those battering rams parked outside your capital."
And again with the Huns. If you can't consolidate the various empires you conquer early on, it's the player's fault not the civs. Once you take so much its easy to consolidate and grow even more powerful for teching after.
No, it's a question of game mechanics. You're taking early-game cities which by their nature have few buildings when you capture them, you're halving their population while others are building up their new cities, and you're losing turns and production to the city's revolt period, while at the same time your early unit focus necessitates that you (a) have few workers to improve the cities immediately, and (b) unless you've made judicious use of bullying, you don't have the gold to buy courthouses (or workers), because such a high proportion of your early gold income is being spent on unit maintenance. It's irrelevant how well you play, since none of these are strategic considerations, they're all consequences of an approach that relies on getting its early expansion through conquest.
And in multiplayer it's the fault of the players of the other civs if the Huns have any kind of advantage. The city positioning advantage is partly a result of spawning, since capitals tend to spawn in or near the best spots in their immediate area, but is also a result of poor city placement by AIs. Human players should be building cities in places that are nearly as good as the sites the Huns capture, and without the early loss of 2-3 pop through conquest and X turns of rebellion. And if the Huns' closest neighbour is - as in your example - Babylon, there is no excuse at all if the Hun rush works. If the Babylonian player hasn't used the time the Huns use to reach their critical techs to get to Archery and Masonry, and then build the relevant units, they deserve to be "outwitted". Yes, if the Babylonian player is a bad player who relies on an inflexible "Babylon must beeline Writing, because that's what Babylon always does" mindset, he'll lose, but the whole point of strategy is that it adapts to the situation you find yourself in. A capital with Walls of Babylon is inherently difficult to take early on - add horse archer-resistant bowmen and the ranged attack bonus from the wall, and the Huns don't have a chance.
If they're up against Persia, they face unique spearmen - tough for the Huns to deal with generally without an equivalent of their own - who are very resistant to archer fire, and require only two techs to reach. Huns lose.
If they're up against Greece, and they aren't the very first civ they hit, the Greeks have Companion Cavalry. I don't even need to elaborate on the effect that has. Huns lose. If they are the first civ they hit, you still have to contend with Hoplites murdering your horses, such that even if you take Athens you won't be moving on to take any other cities in the immediate future.
If they're up against Rome, the three techs it takes to get the Huns underway are enough to get the Romans to Iron Working - it's slightly more hit and miss since that route deprives Rome of early archers and relies on them finding iron. But if they do find iron, or if they're the second or third target (by which time they'll likely also have ballistae) ... Huns lose.
Oh, and one of the reasons I've seen the Huns have so little success in my games? Mostly when they've spawned next to me, I've been playing the Maya. No Hunnic strategy you can name can successfully rush a civ that doesn't need any tech at all to get archers and which can rush-buy them in quantity very early. And to think you're claiming that the Huns getting cheap archers that don't require horses is overpowered... And that's without pointing out that the Maya tech faster earlier than any other civ in the game, so they'll likely have a tech advantage
and defence even if they're the first civ you meet.
etc. etc. True, the Huns will do a lot better early on against civs without early UUs, but everyone can get spearmen and archers before the Huns hit. Everyone can potentially build walls and, horses permitting, horsemen (and often composite bows or swordsmen) before they hit the second time.
And in multiplayer (You all forget this is multiplayer where there is no AI to get obscene advantages)
You forget that the AI gets obscene advantages because it's never going to be as good as all but the weakest human players in terms of strategy or combat. Yes, if you find your Huns playing against people who play exactly like an AI with no advantages, the Huns will probably win. So will any other competent player. It's unlikely the AI would do any of the above because it doesn't have the ability to adapt to the identity of its nearest rival and select the appropriate techs/response to counter it at that stage in the game, yet none of these counters are exactly the height of complex strategy to either come up with or execute.