Phase spectrum range
Yes you can think of phases as describing translations.

If you (kind of passively) wrap the phase to ⟨−π,π⟩ you may be introducing discontinuities that are not truly there considering that −π and π are equivalent. Such artificial discontinuities will hinder calculation of things like group delay.

Unwrapping the phase properly is not trivial. One way to do it properly is to factor the system into such small component systems (connected in series) that you can track the phase frequency response of each without any wrapping ambiguities. The phase frequency response of the composite system will be the sum of those of the component systems. Calculated this way, the phase naturally get values outside of ⟨−π,π⟩. Computationally, the factorization can be done by finding the z-plane zeros of the system. My guess is that numpy.unwrap does nothing of the sort but simply looks for big jumps of the wrapped phase as function of frequency and tries to make those jumps shorter by adding or subtracting 2π. That will easily fail do do a proper job if the frequency response has two zeros at the same frequency, as that gives a jump of 2π that phase unwrapping ought to preserve.