I'm no big U2 fan by any stretch of the imagination, but you're over-underestimating (dunno if thats really a word but I like it!) The Edge there! For one thing, I just looked it up, the chords to Blackbird aren;t really that complex.....the craziest thing is theres a E minor-major 7th in there......maybe the voicings are a bit unusual.....and DADGAD tuning is nothing new.....I think he probably knows of it, even if he doesnt USE it.....
edit:
upon further investigation, it seems he at times used a Dmsus4 tuning (FADDGD).....never heard of that one myself actually......
and I think that should be Dm "add" 4, technically, but that's neither here nor there....
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Blackbird might not be THAT complex in terms of chord structure but it's more complex than much (most?) of U2's output. However, I looked up some of The Edge's tuning variations and he does indeed use alternative tuning for Stuck in a Moment and it's nicely done. I think that's it except for a minor alteration on A Sort of Homecoming.