People go for chinese room for some reason when cartesian theater is the better fit here. What you're doing is placing yourself in the seat of the Homunculus waiting for the show to start. But anatomical investigation reveals that there's no theater at all, and in fact no central system where everything comes together. Instead, the whole design of the brain goes to great pains to tease input signals apart.

Basically, manipulating the symbols won't necessarily have any long term influence on your own state. But the variables you've touched on the paper have changed. Demonstrably; because you've written something down.

If you then act on the result of those calculations, as of course many engineers before you have done, and many after you will do; then you have just executed a functional state change in physical reality, no matter what the ivory tower folks say.

(And that's what the paper is about: Functional states)