As this thread is in the amplifiers section and not the room correction section, I hesitate to post this here. However, it may provide some interesting data for discussion. I myself have been trying to understand phase, and I'm finally just beginning to do so.
I have just added fibreglass acoustic panels to my small office, which has a pair of Neumann KH120s on the desk. And I've taken measurements with and without the panels, as well as a couple of recordings to listen for comparison with headphones (I'm using SRH-1840s which measure extremely smooth & neutral).
At first I smoothed the measurements at 1/12th octave. With smoothing, there isn't a huge difference between the before (red) and after (green) in the frequency response [EDIT: these measurements are for the left speaker only]. The difference in unwrapped phase is more dramatic, and it occurs above the transition band, where room reflections and loudspeaker variation take over from room modes.
I suspect this difference in phase is a result of the reduction in reflections and reverberation in the room, which are audible both in direct listening and in the recordings.
Before:
After:
The reduction in resonances and reverberations is more noticeable in the un-smoothed frequency plots, overlaid here. However, I would still argue that the difference is a little easier to discern in the phase shift plotted above. In any case, a difference is there, albeit differently represented, in both frequency and phase plots.
Whether this has any relevance at all to the discussion is for others more knowledgeable to decide.
Here are the recordings for comparison (performed with a single Sennheiser ME 64 into a Scarlett 2i4):
drive.google.com
Note that I may have also engaged the bass tone control on the speakers between recordings, which was a mistake (although it sounds better), and which explains the difference in the boomy quality of the bass in the "before" recording. However, if you focus on the vocals, you can clearly hear a difference.
[EDIT: the tracks were each normalized to a peak of -3db in Audacity -- and because of the mistake I mention above, and the bass frequencies therefore being louder than the other frequencies, you may need to turn up the volume on the track without treatment to sufficiently hear the difference in the vocals. That said, the better balance in the other track should allow you to turn that one up as loud as you like, without too much discomfort. Not a perfect listening test, unfortunately, but it should be sufficient for a demonstration if played back on a high enough quality system. I can hear the difference clearly on both headphones and speakers.]
Also note that phase shifts are more audible when stereo tracks are summed into mono. So what is audible in a mono recording vis a vis phase may not be audible in person. In other words, a difference is audible in person, but it may not be phase, even though the phase difference may be audible in the recording (I think it is).
And here is the original track: