I hear clicks followed by a tone. But when I reach ~15kHz, the rest becomes clicks followed by silence.
That's a good description. And you have hit the most interesting part - we hear the "click", which is nothing else than the starting transient of the sine tone. There are no lower frequencies in the spectrum than the base frequency. As an example, this is the third sine burst before the end of the test sequence. Frequency little above 19kHz, band limited to 24kHz, starts as a pre/-ringing cluster. We do not hear the base frequency of 19kHz, BUT we do hear the initial transient of the 19kHz (properly bandlimited) wave. In other words, we still hear the transient atack. Put a brickwall LPF 18kHz into to chain and there is nothing audible. This is what I want to say.
So, do we need full bandwidth or not?
. Can we think only in terms of steady state sines or not? How about transients with frequency content above our steady state hearing abilities?