Time/phase only makes relevance if the crossover are active and reside before the speaker drivers.
If the xo is passive and resides after the DSP time/phase delays, the changes will only be applicable to each of the left or right speakers as a whole and cannot impact the individual drivers unless ofcourse one runs single full range drivers.
In a passive environment, the DSP effect that bring is more changes are the equalisers.
All passive speaker manufacturers strive to attain the best possible phase coherence between the left and right speakers through the passive crossover so that one gets the best possible imaging and accurate stage.
Transparency though a different aspect of the design, also aligns within the crossover design.
Without proper tools, it is almost impossible to design a proper passive crossover.
Any pre amp DSP phase/time delays one incorporates in a passive setup will essentially have to do with getting the imaging correct if the left and right speakers are not properly positioned due to room constrains.
Other DSP possibilities include playing with the gain, octaves (equalisers), nowadays even up-sampling or down-sampling signals, preconfigured filters/plugins and even stuff like speaker setup that you find in AV receivers where-in one can measure the distance to listening, back and side walls etc for the DSP to set the delays accordingly (popularly termed room correction).
Some of the most advanced implementation of DSP includes tuning individual tracks to your taste and automatically recalling the same every time the song is played back through the software.
Digital Room Correction (DRC under Soundforge) This is possibly the most powerful and most complex of software DSP implementation under continuous development since 2002. The best part is, it is absolutely open source and free and is supported by JRiver, Foobar and all those playback software that support external plugins.