CR is one thing. Alignment is an other. And fluff is its own thing too.
I don't know how you build your adventures and campaign. But I know that for me, if I want consistency, I will use the standard alignments for my creatures save for very specific individuals (if any). So yes sometimes, alignments might play a big role on the believability of allied races/organisation's. Yes I could change the default alignment, but then I would need to add an other layer of explanation as to why normally non aligned races/organisation's are working together. And here, alignment is not necessarily a hinderance, in fact, it makes it easier for me to spot possible inconsistencies.
With no alignments, again, I would need to ponder upon the written fluff and muse over it. I see alignment as an early warning system that will warn me that if I mix such and such monster type, I will need to put extra work to make things believable.
And fluff is important too.
Example:" I would not put Mindflayers as allies of Duergar, they are mortal enemies even if they share alignment. But I would not put Duergars with Drows either as they do not share alignment. If by some twist of fate I absolutely want to put them together, I know that I will need to knit a tighter back story to make things fit nicely.
Without alignment, it could be easy to put things together with a just because I feel like it. But what happens when a player knows more about the fluff than you and you blunder with an implausible alliance? You get a no way from that player and you might find yourself forced to explain your self afterward and look pretty much stupid for not seeing that. I saw this happen a few times in Role Master, War Hammer and also in Star Wars d6. So yes alignment can be a useful tool in building more believable adventures.