A "shotgun" is by definition a shoulder-fired weapon. A short-barrel shotgun is an NFA shotgun, shoulder-fired with a short barrel. But an AOW made out of a virgin or non-shoulder-stocked shotgun receiver is not a "shotgun" at all, it's a "smooth-bore pistol," a firearm NOT designed or redesigned to be fired from the shoulder. That's the primary difference here and it helps to think of it that way when trying to figure out which to do.
A Remington 870 AOW is an NFA pistol, so of course it cannot have a stock as an AOW nor can it become an AOW if it had a stock as a Title I firearm.
AOWs are cheaper to buy because they transfer on a $5 tax versus a $200 tax for an SBS. As for Form 1ing your own, there is no savings as the tax on a Form 1 is $200 regardless of what you're making.
AOWs can be a loophole in states that don't allow short-barrel shotguns, but this can work in the opposite direction too. Some states use definitions that do not differentiate between an AOW built on a shotgun receiver and a short-barrel shotgun, they partially define it through the ammo it fires or is designed to fire.