When we talk about transient response, the transient is an infinate bandwidth signal. It contains all frequencies. There is no such thing (in terms of transient response) as a transient at 20KHz and a different one at 20Hz.
Yes, no such thing as transient response at a particular frequency.
Personally, i seldom like to see the 'infinite response' card played, with regards to transient response, reproduction of square waves, our FFT measurement programs...etc etc.
Pragmatically, who cares about infinite response other than realizing the limitations finite audio bandwidth imposes on those items?
Imo, transient response with regards to audio, should simply be taken as a given that it applies only to audio bandwidth.
So far, the discussion has mostly been about transient response at a level where the speaker maintains linearity across the frequency spectrum.
About the relationship between freq response and impulse response.
If transient response is meant to as maintaining linear frequency response across the spectrum linearity as SPL is increased, with the linear output needed for headroom....
then I'd say it's 100% that it's an important thing.
Most home gear can't do it, even more powerful gear that supposedly can.
Once you've truly heard it, it goes on the important list..... imho.
IME, when transient response as defined by excellent frequency response (mag and phase),
and transient response as defined by linear output capability,
are both achieved....
opinions change about what's important, rather quickly