Designing a Chirp in the Frequency Domain
Instead of phase delay $td(f)$, use group delay $\tau(f)$:
$$\tau(f) = -\frac{d}{d\omega}\phi(f),\tag{1}$$
calculated as the negative of the derivative of the phase $\phi(f)$ with respect to the angular frequency $\omega$, which is defined by:
$$\omega = 2\pi f\quad\Leftrightarrow\quad f = \frac{\omega}{2\pi}.\tag{2}$$
Using your phase shift $\phi(f)$:
$$\phi(f) = 2\pi n f^2 = \frac{n\omega^2}{2\pi}, \tag{3}$$
we get from Eq. 1:
$$\tau(f) = -\frac{d}{d\omega}\frac{n\omega^2}{2\pi} = \frac{2n\omega}{2\pi} = 2nf.\tag{4}$$
This should explain your empirical results, considering that here $\tau(f) = 2\,td(f)$.