Although I have a lot of respect for Rod Elliott, he should have mentioned that this comment is valid for speaker crossovers, where the significant phase-shift of the recombined signal is not a real issue, but in other applications, particularly multiband compressors, the resultant change in peak factor can be a serious problem.
I have used extensively 4th-order Linkwitz-riley crossovers for sound systems, but for multiband compressors I had to consider different strategies.
I killed way too many brain cells pondering loudspeaker crossovers, the L-R 4 pole alignment is fabulous for high power general purpose sound reinforcement because the steep slopes and -6dB at crossover affords better driver protection against out of passband signal. Back in the early 80s I had some issues making 4 gang alps pot track well enough for crossover frequency adjustment (I had my technician slug them at 50% rotation for better tracking).
Not to get off too far into the weeds but loudspeaker crossovers are dealing with what is effectively a 3 dimensional problem. Not just how the bandpasses sum on axis, but total power into the room, off axis lobing, etc.
That said side-chain dividing circuits are more 2 dimensional situations so the derived filters that sum back to the original signal seem pretty attractive (to me), despite the modest slopes.
JR
PS: Not to veer too much but the best sounding loudspeaker crossover I ever heard was a staggered pole oddball designed by a golden ear engineer in Peavey's transducer group, that had a one pole response through the critical crossover region, with steeper slopes above and below. I gave him a free hand to show me what he could do, and he did.