Why isn't every eigenvalue of a stochastic matrix equal to 1?

I think that you are assuming that the entries of $r$ are non-negative; in general, you cannot assume that the entries of $r$ add to 1, because they might add to zero. Take $$ M=\begin{bmatrix}1/2&1/2\\1/2&1/2\end{bmatrix}. $$ Eigenvalues are $1$ and $0$. The eigenvector for $0$ is $\begin{bmatrix}1\\-1\end{bmatrix}$.

The problem is when you assume WLOG that the entries of $\bf{r}$ sum to one. Consider what happens when entries of $\bf r$ sum to zero.

What your proof shows is that if $(\lambda,\bf{r})$ is an eigenpair, and $\lambda\neq 1$, then the entries of $\bf r$ sum to zero. For example, the matrix $$ \begin{bmatrix} 0&1\\1&0 \end{bmatrix} $$ is stochastic, with eigenvalues $\pm1$. The eigenspace for $+1$ is spanned by $(1/2,1/2)$, and for $-1$ is spanned by $(1/2,-1/2)$, where the latter's entries sum to zero.