Humans have a neocortex, but if you play with a cat or dog, you can recognize and understand the emotions they are feeling. Fear, anger, joy, anxiety, relaxation, etc. That suggests the structures responsible for those things in us are probably not that different from them.
And yet birds don't have a neocortex, and yet they share all the same behaviors that mammals share (at least in some species). I don't know if there is some creature with a neocortex that doesn't share all of these behaviors, but at least we can see that the neocortex may be sufficient but is not required for a species to have them.
IMO the key is distinguishing between these two ideas:
1. There is a gross anatomical structure that can be linked to ancient ancestors with certain characteristics.
2. Those structures in living creatures are somehow "not really modern" or are unusually tied to the needs or limitations of those ancient ancestors.
Consider your fingers: They originate from fin-bones ~380 million years ago, yet (unlike "lizard brain") nobody talks about possessing "fish fingers" except as a fried food product. We also don't create narratives explaining our finger operations or design in terms of what ancient fish required or were capable of.
I am not qualified to argue for or against the Triune brain, but it seems easy to see why the brain is different from the stomach in this regard.