What do you think? Is it important that an input has its time characteristics preserved in the output, or is it all about magnitude of the frequency components?

From listening to the Kii Three's I would certainly argue that there is a subjective advantage in having linear phase, whereas earlier I would have relied on the findings on Linkwitz. What do you think?
What elements, outside of the impact of the recording chain, effect the phase of the audio signal? Let’s make a major assumption that the time component of the original acoustic event is maintained throughout the whole recording chain. Then let’s assume that the electronics powering our DUT is linear. So we have a signal which has maintained the temporal relationship of all signals that is now driving a loudspeaker; how can we destroy the time information?
The loudspeaker itself, especially the cross-over (if any), will affect the phase, and if this is not linear, then there will be temporal distortion. The question is too which degree you can hear this or not...
The literature appears to be very thin concerning the perception of phase information in sound recording. My guess is that very few loudspeakers are capable of accurate reproduction of phase information even when it is present in modern recordings. It doesn’t seem to be that big of a problem.
First listen critically over a premium set of headphones making special note of the aural space. Then listen over a pair of full range speakers (this eliminates the distortion inherent in all crossovers.) Better yet, use full range omnidirectional speakers, preferably ones without enclosures, to eliminate diffractions and reflections. Compare the perceived location of sound sources between the headphones and compare over the omni full range box-less speakers. Use the headphones to evaluate phase accuracy of recordings. Then validate the same over the speakers. This gives the experimenter a reference for subjectivity evaluating the phase components of recordings. Then attempt to correlate observations with measurements of phase in the recording.
I suggest that a subjective analysis of the perception of phase in audio signals should start with the playback of non-manipulated binaural recordings. For this I recommend recordings by Gordon Hempton.
Sound Tracker
https://www.soundtracker.com/about-gordon-hempton/
( Hempton interview https://onbeing.org/programs/gordon-hempton-silence-and-the-presence-of-everything/ )