I am trying to understand the proof In Fulton's and Harris's Representation Theory book where they show that the length of the root is at most 4:
Theorem If $\alpha,\beta$ are roots with $\beta \neq \pm \alpha$, then the $\alpha$ string through $\beta$, i.e. the roots of the form \begin{equation*} \beta - p\alpha, \beta -(p-1)\alpha, \dots, \beta - \alpha, \beta, \beta + \alpha, \beta + 2\alpha, \dots, \beta + q\alpha \end{equation*} has at most four in a string, i.e. $p+q \leq 3$; in addition, $p-q=n_{\beta\alpha}$
where $n_{\beta \alpha}= 2\frac{(\beta,\alpha)}{(\alpha,\alpha)}$ (positive definite form on a vector space $V$) and then the reflection of a root $\beta$ in the hyperplane orthogonal to a root $\alpha$ is given by \begin{equation*} W_{\alpha}(\beta) = \beta - n_{\beta \alpha}\alpha \end{equation*}
They first proof the second part, that is: $p-q=n_{\beta\alpha}$. Then to prove the first part they say: take $p=0$, and then $q=-n_{\beta\alpha}$, which we know is an integer no larger that 3 (a root system here is assumed to be crystallographic) and that's it. I feel really stupid, but I don't understand why is it sufficient to consider the case where $p=0$. Say, $p=1$ then $1=n_{\beta\alpha}+q$. Say $n_{\beta\alpha}=-3$ (am I making a mistake here, considering such case?) then $q=4$ and $p+q=5>3$. What am I not seeing here?
Replace $\beta$ by $\beta' := \beta-p\alpha$. Notice that the $\alpha$-strings through $\beta$ and $\beta'$ are identical sets, in particular have the same length.
You can also check (using linearity of $W_\alpha$ and the definite form $(\cdot , \cdot )$, as well as $W_\alpha(\alpha) =-\alpha$) that once you have the assertion about $W_\alpha$ for $\beta'$, you get it for $\beta$, because e.g. $n_{\beta \alpha} = n_{\beta' \alpha} +2p$, and everything stays consistent.
As to your proposed counterexample: It follows from this theorem that assuming $n_{\beta \alpha} = -3$ enforces, for the reason you outlined, that $p=0$, which case happens exactly iff $\alpha$ and $\beta$ form a basis of a root system of type $G_2$, as seen here.