C'mon Alex:

may I paraphrase your argument:

"After studying many Chess games; we find that Chess moves are prescribed by symmetries between the shape of pieces and mathematical Chess laws.

There is no room for Chess players, all the moves can be shown to be probabilistic deterministic equations."

In so far as physics utilises probability; there is all the room in the world for players.

In fact, thanks to Godel and Paul Cohen; math is rather weak-looking.

In fact, physicists can not fully predict (at micro-level) ANY specific thing.

Freedom rules!