Note that π, like practically any constant or number normally found in mathemathics is constructive and contains a finite amount of information. What the author consider unphysical are the non-constructive reals: unfathomable numbers that can't be defined or written in any finite amount of words or paper. It's hard to argue they play some role in our physical world: it would basically amount to metaphysics.

However, even if rarely used outside of proofs, they are the vast majority of the reals, in fact there are only countably infinite constructive/computable numbers.