Shiite militias are leading the fight against ISIS in Iraq. This is significant for five reasons.
First, the Iraqi Shiite militias are non-governmental forces that the US has neither trained nor equipped. The encouraging success of these groups in combat against ISIS demonstrates that Iraqis can be effective fighters. ISIS believes that Shiites are apostate Muslims who deserve death. This obviously gives Shiite militias a strong motive to fight.
Second, the problems that the Iraqi military experiences – regardless of alleged expert opinions in the US and among senior Iraqi military and government leaders – are not going to be solved by additional training or equipment. The Iraqi armed forces suffer from a lack of commitment and morale among their personnel that no amount of training or equipment can fix.
Third, no matter how much the US or other nations would like Iraq to endure as a unified country Iraq's future is in Iraqi hands. Some Iraqi Sunni groups have joined the fight against ISIS. These Sunni groups fight ISIS for reasons similar to the reasons that Kurds and Shiites fight ISIS: opposition to ISIS' rule and fear of slaughter if ISIS wins.
Fourth, the success of non-governmental forces fighting against ISIS (primarily the Kurdish groups and Shiite militias) puts the problem of ISIS in context. ISIS is not a direct threat to the US or Europe; ISIS poses regional problems best solved by the people of the effected region.
Fifth, ISIS does not justify the US reengaging militarily in Iraq or spending large sums to train and equip Iraqi armed forces. Equipment provided to Iraqi government forces will tend to end up in the hands of non-governmental forces, including ISIS. Air power can aid the fight but is never decisive by itself. Being the wealthiest, most powerful nation does not mean that the US can, much less should, impose its will on other nation states.