You have your parties mixed up I think. You probably meant to write 'UN'. NATO does not 'authorize wars'. Even so, in fact the 'alliance' contained many nations not in NATO:
The really bad part is that they were acting during the 'first installment' against people that weren't part of the attack on the United States to begin with (it was essentially a pretext), and the second time they outright lied about the intelligence regarding the WMDs (of which there already was conflicting data collected by the UN weapons inspectors).
NATO does “authorize” military operations - the NATO ambassadors meet and agree to take military action, which means that NATO assets and staff can be used to help plan and support the action. NATO never authorised the invasion of Iraq, so NATO’s own assets and facilities could not be used. By contrast, Libya and Kosovo were NATO authorised operations, so NATO resources could be used.
NATO’s own resources are essentially office buildings, computer and communication systems, bureaucrats, and seconded military planners. The US didn’t need them for the Iraq War, although it would have liked to have had them involved, because of what that would have meant politically/diplomatically/symbolically.
Despite claiming a broad “coalition”, only three countries actually contributed fighting assets to the initial US invasion - the UK, Poland, and Australia - only the first two are NATO members, and Poland at the time had only recently joined (only 4 years prior). 3 more NATO members (Netherlands, Italy and Spain) provided non-combatant support. I don’t see in what meaningful sense it could be said to be something NATO did.
More NATO members joined in the occupation forces after the initial invasion. But that was never an official NATO operation because those require unanimous approval, and certain member states (most notably France) were always opposed to that.
NATO eventually did become involved in Iraq - training the new Iraqi government’s military. But no fighting ever happened in NATO’s name.
NATO has never said they have to get UN approval to engage in non-defensive military operations. Consider the NATO intervention in Kosovo: NATO member states sought UN Security Council approval for the intervention; when it became clear China and Russia would veto it, NATO went ahead and intervened anyway in the absence of any UN authorisation. NATO still reserves the right to engage in non-defensive military operations without UN authorisation, whenever its member states can unanimously agree that is appropriate.
Yes, and for your and my sake I hope they won't ever have to act upon that reserved right. Because when and if they do I give the world as we know it another 20 minutes or so.
Given Rusdia’s attack on Ukraine, I can easily imagine circumstances in which not acting on that reserved right might lead to the end of the world as we know it.
Yes, unfortunately there are no easy solutions to all this. It will most probably get a lot worse before it will get better again, how much worse I shudder to contemplate.
Oh yes, I almost forgot the "Coalition of the Willing" against the "Axis of Evil". Still sounds like straight of a Marvel movie. Including the Mission Accomplished speak on a carrier, also that part is more like Michael Bay.