Are phase angles in filter transfer functions primarily for stability?
Different applications are sensitive to phase variation with frequency for different reasons.


It can impact stability margin if the filter is in a closed-loop system.
The total error between a source signal and a filtered signal is a function of phase (as an extreme example, if h(x) defines a filter with a gain of 1 and a phase shift of 180 degrees at some frequency, then at that frequency the error signal y = x - h(x) will have an amplitude twice that of x).
If the phase does not change linearly with frequency then the group delay of the filter won't be constant.  This will causes pulses to be smeared out, which is a Bad Thing in communications systems.
If the group delay of a filter is high it will contribute to the overall delay in a communications system.  This is much less likely to be a problem, but can still be an issue in some applications.