Transient of a circuit when a generic audio signal is applied
This is why a step/impulse input is used: it determines the dynamics of the system to a (theoretical) infinite sum of sines. The response can be then extrapolated to any input. Keep in mind that you have to separate the theory from the practise, because you'll also have the bandwidth of the system to help, meaning for (e.g.) an audio amplifier, that sudden piano will be well whithin the bandwidth, and so will be most of the signal, so whatever peak response to step input the system has, it will be much less for the practical case, but it will be there.

Here's a minor example with two 4th order filters: a Chebyshev with fc=20kHz, 1dB ripple, frequency normalized to -3dB, and a similar Bessel (no ripple though...). This is the step response:



The Bessel has very little overshoot, while the Chebyshev has lots and more group delay. And here's an actual audio sample:



If you look, for example, from ~0.4ms to ~0.6ms, the input (black) is one shape, Bessel follows closely the same shape, while Chebyshev distorts it. Also look at the immediately following double peak. The audio sample (.wav file, hence the sharpness) is a kick drum + snare + cymbal, so enough bandwidth.

Conclusion: you're overthinking this a bit. There are more factors to account for, but the most dominant is the linearity of the phase, which is what brings most distortion. I say mostly, because, if it's an analog filter, you have the sensititvity of the elements, opamp, the supply, etc, while for digital you have the dynamic range, while for both cases the noise (internal or external) is inherent. So, the system will repond to whatever input you have based on its own dynamics: if it distorts less, you'll hear less distortion, or none, and vice-versa.