I am not in-tuned with the science but I do agree that one of the most important aspects of a sub woofer is its repeated capability of continuously producing bass notes with how fast it can stop and start.
Although, I have come across many discussions(personally) where experts vehemently deny that anything like fast bass exists even an amateur audiophile understands that wavelengths at that frequency are slow. But there is no doubt that a sub must exist that produces this type of bass and it should be number one on a measurement list to define a subs quality. I know an impulse response gives us a lot but something should be invented in terms of testing for a continuous barrage of repeated bass.
There is a curious sort of mental model that thinks that a sound source like say a bass drum can only be reproduced by a sub-woofer, and so the slam of a bass drum might become muted and "slow" if the sub-woofer is not "fast" enough. But there is nothing special about a bass instrument. All instruments have energy across a wide range of frequencies. Apart from say a flute, musical instruments are awash with harmonics. Violins are a sawtooth, over-driven electric guitar is a square wave at heart. We don't get worried that the mid-bass driver is unable to reproduce the thrash and yowl of these instruments. That is the job of the tweeter. Bass is no different.
Of course integrating the energy from the various drivers isn't trivial. Time delays, and importantly, group delay matter. But once got right that slam you want is going to be there perfectly.
The answer is most certainly not in trying to create some sort of high bandwidth sub-woofer. That is a recipe for all manner of horrors, not the least of which will be making integration with other speakers impossible at a fundamental level.
Impulse response is just one way of representing the response. It is the dual of frequency response. Albeit with some advantages and some problems when used in real life. A continuous barrage of repeated bass won't tell you anything other than the thermal properties of the sub. You will find everything that characterises the sub in that one impulse response. The trick is using the transforms to tease it out into more easily digested form. Taking the fourier transform of an impulse gets you the complex frequency response (which is a fancy way of saying frequency response and phase.) There is enough to get things like group delay and the like. It is all there.