4th order all the way baby.
It seems like most of the high end are 4th order.
Like the KEF Q is 2nd order, the R is 3rd order, and then the Reference, Blade, and Muon are 4th order.
It seems like all the great measuring speakers are 4th order. Must be a good reason for that.
That's nonsense. The correct crossover is the one determined by the driver specs, and what type of lobing and dispersion you want.
Now smooth frequency response is essential. Even small aberrations in the mid band response and even as little as a db is audible.
Fourth order crossovers do to some extent make smooth overal response easier. However just designing a fourth order crossover is pure laziness.
Now we get into phase and time shifts. I do believe you should make the absolute minimum of time and phase shifts commensurate with smooth frequency response and overall deign goals.
I share the view that speakers with fourth order filters, though many are excellent, never go away with the gold.
A good fourth order design compared to an equally good design with less phase and time shift sounds some what "slugged."
Now I admit I'm influenced a lot here by Ted Jordan, who did an awful lot of work on this. There is a degradation in realism, especially the space around instruments, that is harmed the more the harmonics of instruments are separated from their fundamentals, in time and phase. I have certainly noticed this in my endeavors.
Worst case scenarios are when you have to reverse phase a driver to prevent a null at crossover. This does not preclude second order crossovers. The reason is that in the real world the crossover of a two way for instance is seldom at the same frequency for both drivers.
The acoustic orders have to be summed with the electrical orders. So you can avoid nulls at crossover even with second order crossovers.
So I do believe it is desirable to keep phase and time shifts to a minimum, but not as an over riding goal.