Well yeah that is true as far as the linear effects that are present at low volumes are concerned. But in real life spaces, as opposed to a bona fide acoustic lab, there are significant acoustic artifacts that can occur at higher volumes that are mechanically constrained by the springiness of the some fixture, or simply the mass [rest inertia] of some structure when volume is low. Thus, because there is some non-trivial constraint to responding to vibration [or constraing the mode of induced vibration] there can be artifacts at higher volumes that are simply not present at low volumes [not demonstrably extant as opposed to merely not noticeable to human ears].
Without getting totally insane you could end up making signals loud enough to create artifacts not present at modest volumes that are powerful enough to not just make your measurements wonky but be starkly noticeable to the listeners = for example high amplitude signals impinging on: loose bits in spring loaded door hardware, sheet metal parts of light fixtures, wall panels that then flex to the point that they actually rattle against the studs at a loose spot, the cat which then tries to hide in the piano... et cetera.