It has everything to do with it.
Anxious = barking = shock. Dog learns barking cause shock, and overall stress levels drop back to normal. Dog expressed anxiety some other way.
The anxiety argument isn't about barking. "Anxious = barking = shock" Anxiety is totally different than barking. It is like a panic attack, and is more whining, shaking, whelping, none of which can be corrected by an e-collar. How would you correctly time the shock if the dog is in constant panic? It would only add to the overall unpleasant experience of being alone. Separation anxiety is a whole other can of worms, and can not be confused with a dog who simply barks, or chews a slipper while you're gone.
Separation anxiety is a severely misdiagnosed problem. True separation anxiety is very evident. You put on your shoes - the dog starts to whine. You grab your car keys - the dog starts panting and circling. And by the time your dog has built up all of this anxiousness by studying the rituals that he/she associates with you leaving, he/she will go into a panic attack once you leave, bounding at the door, shaking, whining, etc... Thus a shock Collar can not treat the anxiety, as a shock collar at best is a tool used to correct an issue with a well timed stimulation. Timing and stimulation that can't compete with the influx of emotion a dog with separation anxiety has.
The key to solve this is changing your rituals. Your dog has learned the process in which you leave the house, which he negatively associates with being left alone. All of these behaviors become more exaggerated with each repetition of the process. So the only way to truly remove the anxiety is to undue the rituals, and work back to the beginning so that you can build a positive association to you being gone.
Start by putting your shoes on (a ritual your dog might have picked up on as a sign you will leave), but instead sit down for five minutes, and remove the shoes, thus stopping the dog from gaining anxiety in anticipation of step two, which may be grabbing your keys. Repeat this process until the dog shows no reaction (negative association of shoes = to being left alone), and can watch you put on your shoes with no ill affects.
Then move to grabbing your keys (figuratively to whatever your next ritual may be), and so on and so forth, until these procedures no longer cause your dog to be anxious.
After a consistent duration of desensitizing the dog to these "rituals" you are ready to rebuild the foundation in which you can create a positive association to you leaving. This might include saving an extra special toy or bone, and offering to the dog ONLY when you walk out of the door. Keep building positive associations to you leaving, and eventually the dog should be content with the rituals leading up to you leaving, and act of you being gone.
Then if your dog starts to show signs of anxiety to you leaving, or the procedures that proceed you leaving (shoes, keys, etc...) just switch it up, or repeat the steps.
The bark collar will not increase anxiety. To increase anxiety, you must create new stress, confusion, etc. The bark collar does not do that, hence the study.
Are you repressing one of the dog's means of expressing anxiety? Absolutely. Creating new anxiety? No.
I think the key is "expressing anxiety", not actually addressing the anxiety itself. And like I explained, barking is not a definitive sign of real anxiety. And yes, it will create new anxiety, as the point of addressing anxiety is to remove ALL negative associations with the owner leaving.