There are no sodium atoms in citric acid, and losing protons to form a conjugate base won't create them. The citric acid will have to react with sodium already present if you don't add some, which may not be in sufficient quantity to neutralize the acid. Sodium citrate is only one of many citric acid salts, and even that has three varieties, so I hardly think you can say it's "part of the base of citric acid", no?
AFAIK, sodium citrate works by sequestering calcium in insoluble Ca-paracaseinate during the ion-exchange with the emulsifying salt, leaving soluble Na-paracaseinate, a potent emulsifier. Citric acid, though, doesn't have a sodium ion and the amount of citric acid you'd need to separate the calcium would make the cheese sauce too acidic.
> I think it’s important to discern that part of sodium citrate is part of the base of citric acid.
That’s because the base of citric acid is citrate. So when combined into a solution with some kind of sodium you get the effects of creating sodium citrate. As you mention it is a thickener. There is sodium in the cheese you are using as well to help this process hence why you only need a little of sodium bicarbonate to help speed up and complete the break down.
As for why it didnt work for you, I also tried it this morning and had no trouble smoothing out and thickening a basic alfredo sauce. It works great. I now have a bowl of breakfast pasta.
Perhaps you didn’t wait long enough for the reaction to take place or your cheese didn’t have enough sodium.