You're using the triune model to draw some rather lofty conclusions that aren't really up to date with our understanding of neurology. It's way over simplified and doesn't really work that way. More recent studies suggest that the neocortex was already present in even the earliest mammals, so it's not quite as straightforward and the demarcation isn't quite as clear cut, as you seem to be presenting it. "Old brain" doesn't "take over" in the way you're presenting it either but appears to act as a primary driver for those basic functions.

Not sure how to even tackle the loftly conclusions you've made because the don't seem to be built on a solid foundation. I think things might be quite a bit more interesting, and wildly more complex, then you seem to be presenting it. I'll just leave some sources below with a quick note. Not trying to be condescending, or rude, just a topic that is a bit interesting, and a lot of people tend to draw some lofty conclusions from the triune model which has largely fallen by the wayside in neurology.

Read the wiki to see how the model was developed: https://en.m.wikipedia.org/wiki/Triune_brain

A quick introduction to why it was important but has shown to be overly simplified and mostly incorrect: https://medicine.yale.edu/news/yale-medicine-magazine/article/a-theory-abandoned-but-still-compelling/

Further details into how we don't have a "lizard brain": https://thebrainscientist.com/2018/04/11/you-dont-have-a-lizard-brain/

Deacon's paper on rethinking the mammalian brain: https://www.researchgate.net/publication/31439318_Rethinking_Mammalian_Brain_Evolution