In fact, air IS dispersive:
Non-idealized acoustics of real-world sound sources, sound propagation in space; interaction with the environment; the ubiquity of naturally occurring sound modulation
hearingtheory.org
Fig. 3.3: between 100 Hz and 10 kHz there is a difference of about 0.06 m/s, which is roughly equal to a phase rotation of 4° at 20 °C, 50% humidity and at 2 meters from the source.
But the temperature in a room is not homogeneous. In fact you can have strong gradients between opposite surfaces (e.g. between ad external and internal wall), and we saw that a difference of only 1 degree can lead to a phase rotation of the magnitude of the phase distorsion of your amp. And here the problem is interesting: low frequencies do not see the gradient but high frequencies do.
And I did not mention the air reactance near the source of the walls.
In my opinion, speaking a perfectly flat phase is rather vain.