..."This chaos and mayhem in Iraq started because of [the] decision to dissolve the Army." ...
The veracity of this statement is not blindingly obvious.
Play a little "what if" with me ...
Suppose the US had NOT sent these Sunnis home.
Would the Sunni insurgency still have taken place? You can make a good case that that it still would have occurred, because all of the government officials were Sunni, not just the Army.
Would the Sunni army have attacked Sunni insurgents? Given the tribal basis of their society, you can make a good case that they wouldn't have ... or they would have warned the insurgents in advance of every movement.
Would the Shias have participated in the creation of a new government, where the Army was still Sunni? Remember, this Sunni army was the one that carried out Saddam's orders to use WMDs on the Kurds, and the one that slaughtered the Shias when they revolted after the 1st Gulf war. If not, then the Shias would be the leaders of the insurgency, with assistance from Iran.
Would the Shias have joined an Iraq army officered by the Sunnis? I don't think so, and so the Shias (e.g., As Sadr's army) would have joined the insurgency. I would much rather fight a minority (Sunnis) than the majority (Shias) in Iraq. Disbanding the Sunni Army was the price for not having a Sunni insurgency.
So ... I believe that the US carried out the least damaging strategy -- create a Shia-majority government and a Shia-majority army, then bring in some Sunni officers later.
umm, no.
Allawi brought back a good percentage of the Army. He simply reactivated old units. And, they were fiercely loyal to him.
Why? Because he was paying them well and they liked him personally.
The Iraqi Army would be loyal to anyone who is putting money in their pocket and they can somewhat stand.