Group Delay for Hilbert Transformer and Resulting Dispersion
A real-valued system that doesn't distort the shape of the input signal must have the following input-output relation:

$$y(t)=Ax(t-t_0)\tag{1}$$

with arbitrary real-valued constants $A&gt;0$ and $t_0$.

In the frequency domain, Eq. $(1)$ corresponds to

$$Y(\omega)=Ae^{-j\omega t_0}X(\omega)\tag{2}$$

Consequently, the corresponding system is an LTI system with frequency response

$$H(\omega)=Ae^{-j\omega t_0}\tag{3}$$

i.e., the magnitude response must be constant and the phase response must be linear:

$$\phi(\omega)=-\omega t_0\tag{4}$$

Here, linear is meant in the narrow sense of the word, i.e., it must not have an additive constant term (hence, it must not be affine). Note that for real-valued systems, the phase must be  an odd function of frequency, so the sign of a possible additive constant in the phase would need to change depending on the sign of the frequency. Consequently, if there's such a constant phase term (as is the case for a Hilbert transformer) then the phase has a discontinuity at $\omega=0$, which implies that the group delay is undefined / infinite at $\omega=0$.

The phase response of an ideal Hilbert transformer with a possibly non-zero delay $t_0$ does not satisfy $(4)$ because it is given by

$$\phi(\omega)=-\frac{\pi}{2}\textrm{sgn}(\omega)-\omega t_0\tag{5}$$

As mentioned above, the phase of an ideal Hilbert transform has a discontinuity at $\omega=0$.