Split brains are incredibly interesting.
AK is right, while the Corpus Callosum is the main highway between hemispheres, there are backroads too, and new paths can be adapted with neuroplasticity. I can't remember the details (it's been a while).
As a general rule, the brain can be incredibly adaptive (e.g. it can remap sensory processing of one region to another within minutes of losing a limb) and will kind of just make up a solution. While it's true that motor control is contralateral, it's actually a lot more complicated than that (for instance, the inhibition for motor movement is a bilateral endeavor).
So rather than thinking of it as a static control system, it's probably better to think of it as a society of neurons, where the very structure of the system can shift and adapt to meet different needs. If you cut off the internet yes communication between geographic regions would be hurt, but there are other, less efficient ways that we would get by.
I would be very hesitant to suggest that this is evidence for the soul. There aren't many better ways to come to reject the notion of a soul than to study neuroscience. I think your curiosity is admirable and your questions are good, I just don't think they'll lead you to where you want to go. Just remember: Everything you hear about the brain from sci articles or docos is going to be extremely low-resolution information in order to make it even a little bit accessible. Neuroscience is a pile of seeming contradictions that take enormous time and effort to parse due to the complexity of the system they're nested in.