In a 2-complex $X$, it is "obvious" that the simple closed curves through the $0$-cell $v$ generate the fundamental group $\pi_1(X, v)$. (By a "simple closed curve" I mean a loop which does not self-intersect.)
Do you have a reference for this which I can cite? Hatcher does not prove this, so far as I can see.
Some obvious things are less obvious than others....
Let $X$ be a rank two graph consisting of a disjoint pair of circles plus an arc with one endpoint on each circle. Then no matter what $v \in X$ you pick, $\pi_1(X,v)$ is not generated by simple closed curves through $v$.