It is certainly a good idea, but it is not always easy to determine what people believe. On the present topic, they believe quite a variety of things, many of them extremely unclear. Why do you assume that if brains implemented soul-code as you describe, someone would think it was the brain that was them, rather than the soul-code? The view that brains implement soul-code and that the soul-code is the real person sounds to me like a not implausible interpretation of much of what Plato says about souls.
No plot spoilers here, just wanted to flag a bit of poor reasoning that shows up in Chapter 39:
This is a surprisingly common fallacy. Just because X depends on Y, it doesn't follow that X depends on nothing but Y. A phenomenon may involve more than just its most obvious failure point.
To illustrate: Suppose I'm trapped in a box, and my only way to communicate with the outside world is via radio communication. Someone on the other end argues that I don't really exist -- "There's no person beyond the radio receiver, for if there was then there wouldn't be any such thing as damaged radios!" Pretty silly, huh? But people say this kind of thing in defense of physicalism all the time.
(N.B. This is not to defend the existence of souls. It's just to point out that this particular argument against them is invalid.)