When I'm dealing with an upset customer on the phone, disarming means that we can move past the "I'm pissed at you, why can't you fix your stuff" useless conversation into the "how can we help you" much faster.
In other words, disarm means the "emotional" center of the brain can safely disengage and the logical center can take over. When someone is angry, most likely, their emotions have taken over and, once that happens, you're going to spend a lot of time dealing with emotions and not moving toward a mutual solution.
Therefore, if you look at it this way, how is the pure emotional response from the aggrieved helping them get to a sort of recompense? I would argue the short answer is that it is not helping at all, it's just pure lashing out. Which, of course, could be a rational response to the situation. However, no matter how justified the aggrieved may feel in lashing out, the act of lashing out itself doesn't actually move the needle.
If I can show compassion and empathy toward the customer, including saying the simple words "I'm sorry" I'm trying to move out of the emotional state and into a logical state where we can come to some sort of agreement on the next course of action. I'm not trying to "get out" of anything, just trying to literally disengage a part of the brain that is not doing anyone any good for either party.
To your point though the apology has to be authentic. And, yes, I do empathize with the customer's anger in the moment. I've been there myself. So I can easily put myself in their shoes. Of course, as the great philosopher Daniel Tiger says, "Saying I'm sorry is the first step, now how can I help?" [0]. So unless you just want to enrage the customer further, you need to immediately follow that apology up with action - in my case, it could be escalating to the right engineering group, finding a workaround, helping identify the root cause, whatever it could be...
Hope this helps clarify. I'm in no way trying to "pull one over" anyone. I truly want to help - which is probably why it works so well.
[0] https://pbskids.org/videos/watch/saying-im-sorry-is-the-firs...