A real-valued system that doesn't distort the shape of the input signal must have the following input-output relation:
with arbitrary real-valued constants $A>0$ and $t_0$.
In the frequency domain, Eq. $(1)$ corresponds to
Consequently, the corresponding system is an LTI system ...
For a phase distortion metric I recommend using “group delay variation”. The definition of Group Delay is the negative derivative of phase with respect to frequency. The Group Delay is the delay in time that “group” of signals over a band of frequencies would have. A frequency response that is linear in phase (constant group delay with no variation) is not ...