Daniel Dennett, in his refutation of Lucas' argument (which I think is in his book Brainstorms; I have not been able to find a version of it online), makes a point related to your point here in a very stark manner: by showing that, in fact, a physical realization of a formal system can produce a proof of that sytem's own Godel sentence! Dennett's argument makes use of a critical observation: in order to view any physical system as a realization of a formal system, we have to interpret the inputs and outputs of the physical system in a particular way. But there is nothing in the laws of physics that rules out the possibility of multiple interpretations of the same inputs and outputs. So it is perfectly possible for a given set of outputs of some physical system to have one interpretation as sentences of formal system A, but also another interpretation as sentences of formal system B, with the latter sentences including a proof of the Godel sentence of formal system A.

Put abstractly this way, this seems unlikely, but in fact we experience this sort of thing all the time with computers. As I am typing these words, my computer is taking a series of inputs and producing a series of outputs which have multiple interpretations: as sequences of bytes, as machine instructions, as typed characters from the keyboard and displayed characters on the screen, as changes in voltage across transistors, etc. And there is nothing in principle that prevents, for example, writing a program that runs on my computer's CPU and proves the Godel sentence of that CPU, considered as a formal system. Or writing a program in, say, Python that proves the Godel sentence of the gcc compiler used to compile that instance of the Python interpreter, considered as a formal system.

So in fact the author of this article is not just assuming that any machine will consist of a formal symbolic system, he is assuming that there is one unique interpretation of any physical machine as a formal symbolic system. And we now have abundant evidence that that assumption is false.