We don't understand enough about thinking/consciousness to reduce them entirely to physics (the literal brain).
> The answer to this is that, upon more careful introspection, it seems I don't even really feel like I have free will either.
Why did you write the above reply then? I posit that I have free will because I perceive my typing this as occurring due to my free will. Maybe everything is (effectively) deterministic, but at the very least I exist in a reality where I can think and act in ways that demonstrate free will. That's more fundamental than any scientific finding, which is necessarily an approximate view of reality. Think of a computer that can somehow emulate computers such that apps in the emulated computers have no way to discover a distinction (timing and side effects are emulated too, I guess). What difference does it make to those apps that they're running in an emulated computer or the base?