Phase response of $H(f)=e^{-j2{\pi}ft_0}$
Wouldn't it be much easier to just compare the left-hand side and the right-hand side of

$$e^{-j2\pi ft_0}=|H(f)|e^{j\phi(f)}\tag{1}$$

to see that $|H(f)|=1$ and $\phi(f)=-2\pi ft_0$?