You are conflating two things. Barb rush and AI civ rush. Barb rush is nothing. I can beat back a Barb rush while still moving forward with my agenda because Barbs can't capture the capital. I have done this countless times no problem. The only time a Barb rush is a problem is if you get hit by 2 at once in that time frame (which will make it almost impossible to kill them all before you need to start doing other things - if only one barb camp is attacking my capital, I am almost certainly going to kill them all before it is necessary). That has happened a few times, but again, it is relatively rare. Maybe 5 - 10% of the time in my opinion. But once again, if you get rushed by 2 barb camps at turn 10-14 it doesn't matter which civ you are playing with. You are going to be crippled with your development. Playing as Aztec is not going to magically fix that vs Nubia.
If you get hit by a Barb rush and an AI civ attacks, that is actually usually a win for you. Barbs will attack ANY units that are within range of them. Thus, if AI civ attacks you while Barbs are near your city, the Barbs will act as a bonus defense force and will attack the AI civ units. Might even kill them, but will certianly weaken them enough for you to finish them off. (Also, note that the AI Civs will prioritize killing Barbs over almost anything. I have seen the AI civ make some very stupid decisions in the past in order to attack a Barb unit, so the Barbs will likely be a very useful distraction in addition to killing the AI civ units.) Plus, the way that Barbs work means that they will chase after retreating AI civ units. So if AI civ declared war and has units in your area, and then they get damaged by Barbs and try to run away, the Barbs will leave your city alone and chase them back to their city!! This is a win for you!
The only way this is a loss is if the AI civ times their attack so perfectly that they attack your capital after you have killed all the Barbs, while your city is still heavily damaged, and while you somehow did not build an army to kill the Barbs. We might be talking about a 2 to 3 turn window assuming you decided to kill all the barbs with one or 2 units. I don't know that that has ever happened to me.
So I don't think you should be counting Barb + AI civ rush as one of the problems, because most of the time that combination is not a problem. And a single Barb camp rush is usually something that can be taken care of fairly easily. The only way it is a serious problem is if you get rushed by 2 Barb camps at once or an AI civ attacks you by itself with a big group of warriors, all between tuns 10 - 14, and if that happens you are basically screwed no matter which Civ you are playing.