Actually emotions and rationality in the prefrontal lobe can work together (see Damasio's classic work on this subject).
Metacognition theories do introduce the idea of an "affective regulatory loop" that utilises *emotion* for executive functioning.
And so on.
What makes you think that people are all the same in terms of "cold objective" rationality and differ in terms of feelings? That to me didn't sound all that logicalStereotypes of "ENTP" and "ISFJ" can exist. But it's doubtful that that's how they act all of the time. They may change their minds, they may change over time, they may learn new things that make them see things in different ways. The most important of all, it doesn't explain why they're behaving that way.
My personal view is that people don't differ in their ability to be logical and rational. But some people may have less feelings, or even are just less aware of their own feelings and the feelings of others.
I would say the easiest way to refute this idea is think of the stereotypical male vs stereotypical female. Yes those are stereotypes and how your brain works doesn't depend all that strictly on whether you have a penis or a vagina but there are actual differences in the brains of people regarding "masculine" and "feminine" stuff. Ofc it's an individual mix of different skills/traits for everyone.
The socionics model does attempt to explain things, it's another issue altogether that the explanations are not holding up.
What @Heretic 007 said is an ok way to look at it. So no don't expect an actual explanation beyond what that can offer - though in my opinion even those explanations do not hold up well.