This is not a rebuttal to the myth. Imagine I'm talking about a car, and I say: It's a myth that specific parts of the car have specific functions. For example, the engine is involved in starting the car, accelerating the car, decelerating the car, regulating the car's speed, producing heat for the cabin, and running the alternator to produce electricity for the lights, power systems, and radio! That's so many things! But of course the engine has one dedicated, primary function, and the others are either downstream functions or side benefits. The anterior cingulate cortex is clearly not as specialized as a car engine, but it is highly specialized; it's just that its specialized function is not so easy to describe (or even discern) in words like "memory", "emotion", etc. Each of those words describes a huge group of functions that are downstream of the function of the anterior cingulate cortex and many other specialized structures.
I believe the "specialized brain regions" idea has been over-debunked. It was the source of so much woo woo in the late 20th century (are you right-brained or left-brained!?) that we've come to think it's complete bunk. But we have a huge body of evidence showing that there is a big difference in how the left vs. right brain processes information and that different brain regions are highly specialized, but that their specialized functions don't map cleanly into the language we were already describing human behavior with.