How universal is the trinity in myths globally? It’s an important part of some flavors of Christianity, but e.g. Judaism, polytheistic religions, Traditional Asian belief systems like Taoism or Confucianism, etc. to my knowledge don’t have the same kind of structure. If it’s derived from the human universal of our complex brains, we should expect to find it in all human cultures. And human-like brains (at least to the extent they are triune in the way you describe) are present in non-human primates. It’s tough to test for chimps’ reverence for the number three, but at least it suggests that any influence that had on our mythology should be present long before the advent of Christianity (or rather, the Catholic dogma of the trinity, which developed much later as far as I’m aware).
I also think the number three is special in ways that can be appreciated by any intelligent being, regardless of the number of layers in their information processing system. In fact, I would hypothesize that the causal arrow runs the other direction: the number of brain layers we have is due to the specialness of the number three. A three-body system has a particular set of complex dynamics, and it’s possible that it’s the simplest system that permits a certain kind of high-level infromation processing. Indeed, see a similar dynamic arises in computer design:
I admit it’s hard to trace causality here; other diagrams omit hardware and show kernel, libraries, and applications as the three layers, so it could well be that we make three-layered computer architecture diagrams because we like the number three, rather than because there’s an inherent three-ness to information processing.