How to apply the substitution method to n/2?
You're entirely correct - if all you prove is $T(1)$ and $T(n/2) \Rightarrow T(n)$, then you prove that $T(n)$ holds for all powers of $2$. One way to fix that is to prove $T(n)$ for all odd $n$, but that defeats the idea of induction. Instead, usually one actually proves the stronger $T(\lfloor n/2 \rfloor) \Rightarrow T(n)$ or $T(\lceil n/2 \rceil) \Rightarrow T(n)$, and both are enough to deduce $T(n)$ for all $n$.

In many cases, if you can prove $T(n/2) \Rightarrow T(n)$ then you can also prove the stronger claims, but it's technically more awkward. So instead of dealing with this mess, you are only shown the weaker claim. The "interesting" ideas are found already there, and to prove the stronger claim requires just working harder but no new ideas (other than routine tricks used for handling floors and ceilings).