The radio analogy is bad because of the first person point of view.
The radio == the external behavior of the person, and you == the soul, right?
Damaging the brain doesn't just affect the external behavior of the person, it also affects the person's thinking, i.e. the person itself. Therefore, the brain isn't just a receptor for a signal, as the radio is, it's the thing doing the thinking.
That's certainly true, but I'm not sure it matters.
One could replace the radio receiver/transmitter analogy with an algorithm-specification/algorithm-implementation analogy, for example, without significantly affecting the argument. Is the hardware the thing doing the thinking? Well, yes, in that when we destroy the hardware we prevent the thinking. But, no, in that even after we destroy the hardware we can recover the thinking by implementing the algorithm on other hardware.
Generally speaking, I think it's better to address the argument being implied by a metaphor than to address weaknesses of the metaphor itself. See also "steel-manning."
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.)