Generally, which shotgun build you use depends on how much you're willing to depend on your shotties. MAD is super-conservative in that respect: armor-piercing rounds turn scouting into a way to kill things, and double shotty WILL kill an up-close spider or rev in one shot, with few exceptions (super shotgun turns that into no exception). MFa, on the other hand, is a matter of being able to use the double shotgun whenever the hell you feel like it, which allows you to conserve shells more often, thus lowering your need for shell space, thus giving you room for other guns. If it's AoSg, I would always recommend MAD over MFa, especially if you want to reach Shotty Diamond.
As per your other comments:
Angel of Light Travel:
Due to inventory being scarce, Fireangel is a lot better since you can get by with less ammo.
Some challenge are really better off with neither build, and AoLT is probably the most Ammochain-reliant challenge I have ever seen. If you really had to play AoLT without Ammochain, I'd STILL probably choose super-modding a chaingun because it carries the most ammo.
Angel of Purity:
In all of these, health is at a premium, so you'll want TaN and Badass.
On the contrary, Purity makes Badass practically useless because you'll never go past 100%! However, MFa still allows for TaN, so you can try TaN->TaN->HR->HR->Rel->Rel->SM->DM->MFa, or something along those lines.
Angel of Max Carnage:
Doesn't this challenge make Dodgemaster useless? In any case, you will need to focus on killing demons before they kill you, rather than escape. This makes Army of the Dead an obvious choice if we're only talking shotgun master traits--although Ammochain is probably even better.
To answer your question: no, to my knowledge, DM still does exactly what it does, even in a game where to-hit bonuses are maximized. Remember, DM AUTOMATICALLY dodges all attacks on the first sidestep. I should point out, though, that enemies can still miss you during AoMC when it comes to dodging an attack, so never forget to run in a pinch.
With that in mind, MFa is useful when dodging the occasional projectile you'll allow to fly, and the already-maximized damage you're dealing with shotties doesn't make me worry all that much about armor. It'll take some more shells than MAD, but (and again, you can get TaN at any point) avoiding splash damage can be absolutely critical.
Angel of Pacifism:
Army of the Dead is useless, obviously. Fireangel is useful, but is it really worth pouring three experience levels into useless traits? You're probably better off just going for Dodgemaster and Badass, using your extra levels to build Hellrunner, TaN, and Ironman.
Depends, is avoiding splash damage that useful? Yes. Yes it is. However, it doesn't really matter for the Diamond run when you aren't allowed HR, which has now been made harder by the fact that MFa requires HR. If it doesn't matter, then I would probably like MFa if only because splash damage is a lot of the endgame damage for a person who can't attack anything.
EDIT:
On another petty note, I don't even like to take Shottyman when I'm playing AoSh, much less any other time.
I'm curious is to what you'd get if not Shottyman for a shotty run. SoaB is remarkably insignificant damage for any shotgun (double included) and Finesse, should you be correctly using the combat shotty, is only useful once the double comes into play...which it doesn't until barons, taking you to around clvl 5 or 6. The rest are either defensive or useless for shotguns.
If you refer to the trait that Shottyman GIVES...well, perhaps it's not as useful as some of the other adv traits (Badass's 200% perm, Intuition's monster vision, Whizkid's super-modding), but it's remarkably useful if your timing's right: fire, sidestep, and repeat, never worrying to reload. Combined with HR (ala MFa) it makes reloading REALLY fast and almost impossible for you to get hurt. Timing of the enemies, however, is critical and may or may not require Finesse to be optimized. Don't expect to pull this off against Arach caves, for instance.
If you, however, refer to the fact that you need two Reloaders to get Shottyman, making either one or the other kinda useless...yeah, I tend to agree. At least Reloader can still useful on a non-shotty build such as a non-MGK pistol game or a non-MAc damage run (things like Railgun and rocket launcher really benefit).