Problem in Papadimitriou's "Computational Complexity" seems odd
Rice's theorem is about non-trivial properties of partial functions. It states that if $P$ is a non-trivial property of partial functions, then no algorithm can decide, given the description of a Turing machine $M$, whether $L(M) \in P$.

Rice's theorem doesn't say anything about the operation of Turing machines. A property like "$M$ ever writes $\sigma$" is not a property of $L(M)$, and so Rice's theorem doesn't apply to it.