Help me understand the phase of an FIR filter
Your answer does not take into account the possible sign of the magnitude term in the polar description of the frequency response $H(\theta) = A(\theta)e^{j\phi(\theta)}$. I guess you mix this definition with the more usual definiton of  $H(\theta) = |A(\theta)|e^{j\phi(\theta)}$ where the magnitude will always be positive and therefore phase will be what you predicted, however for your example $A(\theta)$ may assume a negative value and you shall reflect this as an addition of $\pi$ into the phase where $e^{j(\theta + \pi)} = -e^{j\theta} $