I think you are just wrong, and you do not seem to know what flat frequency response looks like in impulse response graph. It is one sample long singular spike, followed by perfect silence afterwards, forever. You could say that studio monitor speaker systems strive to reproduce just such an impulse, and the closer it is to a very narrow spike, the better the acoustic system. Even this headset looks like it is not that far from perfect impulse apart from some ringing afterwards which suggests it has some resonance peaks and probably highpass filtering because the impulse goes below zero, and that would indicate it cancels some of the sound it produced earlier after a time delay, which is how highpass filters generally work. I don't know, it is really hard to try to read the frequency response off impulse response.

It is completely obvious to me that time delays do not have impact on the magnitude spectrum. They have an effect on the complex spectrum because phase (and group delay) are different and are encoded in the ratio of the imaginary and real parts of the complex number which is usually not shown because phase angle is difficult to relate to anything we actually hear. In that case, group delay would show the added fixed delay just fine, though.

I can only assure you that from mathematical point of view, the impulse response and complex frequency response can be converted to each other without loss. Whether Fourier analysis is good model of human hearing is perhaps a thornier question, as this isn't quite how our ears work, but I think it is still plenty useful as a construct.