We have this idea that we each have a single, unified “soul” or “mind”, but what we think of as our conscious experience really contains a lot of systems that interact and sometimes conflict with each other.

Very true, and to add to this there's a fair bit of research suggesting that our conscious awareness (in the moment, at least. Harder to design experiments for long term planning, learning, and behavior changes.) is more of an apologist for deeper neural decision-making processes, separately perceiving what we seem to be doing and creating rationalizations out of whole cloth for what we're doing and why we're doing it. I don't have references off-hand, but I'm thinking of studies on split-brain subjects (subjects who have had the main structure connecting the right and left hemispheres of the brain severed, often as a last resort procedure for severe epilepsy or sometimes via brain injury) and studies investigating brain signals that immediately precede motor behavior.

If there’s a movement to make it more mainstream to talk about human minds like this and to develop a vocabulary for talking about ourselves that goes beyond treating people as singular, unified selves, that might be a positive thing.

That would be a very cool and interesting result.