Interpret group delay of a filter
This is actually pretty simple. Take a pure-tone sine wave, say

$$s(t) = \cos\left(\omega t + \phi \right).$$

Now, fix a time $t_0$ and find the phase $\psi(t_0)$ of $s(t_0)$. We get

$$\psi(t_0) = \omega t_0 + \phi$$

Now let's vary the frequency $\omega$ and keep the time fixed. The above expression becomes a linear function in $\omega$! Even though we haven't actually "delayed" the wave.

Imagine it visually. You have a point in time (not 0, that will always stay the same) and a sine wave which is, say, shrinking (frequency increases). As it shrinks, different phases of this sine wave will cross your fixed time point for different frequencies.

This is why phase response of a simple delay system is a sloped line, not a horizontal line. Its derivative, however, will be horizontal and proportional to the amount of delay we introduced in the system, i.e

$$\frac{\partial}{\partial \omega} \left( \omega t_0 + \phi \right) = t_0,$$

Which is precisely the system delay.