I wouldn't say that the Triune brain theory is a 'myth'. It's just not accurate, but it's not a complete lie either. There certainly are different parts of the brain that have evolved at different times, and have different functions.
The older parts have evolved to specific functions, while neocortex, which is found in mammals, has a uniform structure and is capable of learning and adapting to different tasks. 75% of human brain is neocortex.
The article also has this weird notion that the only source of evidence of the Triune brain theory are brain scans. We also have evidence Wilder Penfield's work stimulating parts of the brain. We also have evidence from damage to the brain, from physical injury, lesions, tumors, surgeries, strokes, etc.
While that data shows that brain functions aren't strictly fixed and neuroplasticity is a thing, it does show that locations in the brain do have disproportionate impact on certain functions.
Barrett's views are not the only ones there are, and they are still debated in the field, however much she may insist to the media that her view is the only correct one and that all else are myths. Things are not nearly as settled as she claims.
Panksepp, J. (2007). Neurologizing the Psychology of Affects: How Appraisal-Based Constructivism and Basic Emotion Theory Can Coexist. Perspectives on Psychological Science, 2(3), 281–296. https://doi.org/10.1111/j.1745-6916.2007.00045.x
Panksepp, J. (2008). Cognitive Conceptualism—Where Have All the Affects Gone? Additional Corrections for Barrett et al. (2007). Perspectives on Psychological Science, 3(4), 305–308. https://doi.org/10.1111/j.1745-6924.2008.00081.x
Yea, I think that in some ways the “myths” presented can be seen as abstractions that lack details for the sake of being digestible. It is also a fair point that these abstractions can be poorly carried to detailed conclusions like the example of humans being essentially more evolved; I think that’s the bone to pick
This article takes obvious simplifications/generalizations and tries to act like it's all wrong because they're simplifications/generalizations. Obviously simplifications leave out detailed minutiae, that's the point. It doesn't mean the general concept is wrong, it just means that there's more to it once you get into the details.
I'm not an expert on the topic, but as Jeff Hawkings says in the video, the differences are caused by the fact that different areas of neocortex are connected to different inputs. The underlying structure is uniform, which can adapt to different inputs.
The older parts have evolved to specific functions, while neocortex, which is found in mammals, has a uniform structure and is capable of learning and adapting to different tasks. 75% of human brain is neocortex.
Jeff Hawkings explains the difference between the 'old parts' and 'new parts' pretty well in this interview: https://www.youtube.com/watch?v=-EVqrDlAqYo&t=366s