Let's put it this way: we know for a fact that what you feel and think can be influenced through electrical, chemical and mechanical interventions to the brain. Given the extent of the "circuit bending" capability that medical science has observed, it's hard to think of a mind as something other than just the runtime state of the brain that hosts it.
There’s a deep flaw in your reasoning, in that it implicitly relies on its own conclusion. To see why, consider an analogy:
Tampering with a radio produces artifacts in the music playing, and can even cause it to change between a number of different signals, but the source of the music signal, the composition of it, is largely remote to the radio.
Your analogy is missing the fact that a brain is a bidirectional device. It has both outputs and inputs, and the two are visibly correlated.
To fix the analogy, consider circuit-bending a handheld transceiver. Assuming the "thinking" / "free will" part represents the conversational partner on the other side of the radio link, modifying my transceiver can alter not just how I hear my partner's transmission, but what my partner receives from me as well. Being able to modify the input, I can influence what my partner will say next.
The brain has a lot of inputs (a very important sense group is the so-called "interoception" - the sense of your internal body state), all of which send telemetry completely under the radar of the conscious mind. Only few of those signals surface explicitly for a mind to introspect, but the rest still influences how the mind thinks. For instance, it may take many minutes before you realize your suddenly irrational behaviour is just a sign of hunger.
My conclusion is this: even if you believe (against Occam's razor) in the mind being a separate, remote phenomenon, and the brain being just a transceiver of some information carrier we have no first clue how to detect - the brain still seems to be the single point that collects the inputs and outputs of that communication. Essentially, it could be seen as an electrochemical I/O module, connected to the mind via Magic waves. There is zero functional difference between this model and (simplified, Occam-friendly) the model in which there is no remote mind, and the brain is just the whole computer.
(And, unless you can show that "remote mind" vs. "brain mind" have different testable predictions, there's zero basis to believe in the "remote mind" version, no matter how much more philosophically comfortable it seems.)
The equivalence between those two statements seems to fall apart in the case of a remote “you”, because we can meaningfully distinguish between your abstract intent and various physical impulses I’m not sure we can in the locally physical model — eg, mixing of your “true” signal with a secondary one because the addiction structure adjusts your tuning; addiction would then be your body no longer accurately carrying out your will, which is distinct from you wanting them.
This isn't a disproof. Just as the radio is an instrument that plays music from the outside environment, a brain is an instrument that behaves in response to the outside environment.
The existence of radios shows that being able to interfere with a signal locally via one mechanism (interfering with the radio electronics) doesn’t entail it’s generated locally by that process — the local process can merely be translating between media, eg light and sound.
Unfortunately, this pokes a hole into the logic that thinking is necessarily local, because it was depending on clinicians interfering locally to show that brains are a local phenomena — but that reasoning doesn’t apply generally, eg, in the case of radios.
You’re correct that we know the brain is going to be doing some kind of transducing, but that doesn’t mean it’s doing the thinking — it could merely be a relaying device between whatever generates thought and the body, like a radio translating between EM and sound.
