The fourth dimension of the universe is time. Passive crossovers have built in time delay, in proportion to the order of the crossover. So a fourth order crossover delays the woofer a whole cycle behind the tweeter in a two way, and two cycles behind the tweeter in a three way. It has been shown that more than a quarter wavelength time shift is a problem. That is why first order crossovers have had their devotees. The problem is that first order crossovers have huge overlap, and frequently put drivers out of their design bandwidth. Now you won't see any of this in standard loudspeaker measurements. But I can tell you that if you drive a speaker with fourth order crossovers with a square wave and record the trace from a microphone, what you see is pretty much a perfect sine wave!
In fact you can use delay to the point where an FR looks perfect, but the speaker is useless and human speech unintelligible. So this speaks to transient response.
This effect is all the more serious as you lower the frequency of the crossover, this is just one of the reasons I will never design a passive crossover below 400 Hz at the lowest, and by the way nor should anybody else.
With active crossovers and now with DSP added, time alignment at crossover is achievable. This is a huge advance. So I think in the near future transient response will feature in measurements, and I expect square wave responses will be included going forward. Right now a speaker's square wave would shock you. The only speaker that has reproduced a reasonable square wave is the Quad ESL63. Two out of phase can cancel a square wave in mid air.