Having followed this campaign since 2014 closely, daily I can say that there's been times the US was idle and let US advance whilst they could have stopped them. A very clear example of this is the fall of Ramadi in 2015, where a week before its fall previous US chief of staff Martin Dempsey stated in a television interview that 'Ramadi is not an interview, they can reclaim it later'. What followed was a massive IS convoy entering Ramadi as seen in the image below, the USAF could have leveled this.
IS entering Ramadi
Until a point all serious USAF support was only for Kurds, that only changed once Russia began air operations in Syria. US-led coalition air support become more intense for the ISF which increased gradually, Mosul air support was maximum. IS could have been defeated quicker if not for all the political obstacles and strings that slowed down the process and costed many life's as well.
Without US involvement Arbil would have fallen to IS and the ISF would not have reclaimed Mosul yet, Baghdad would not have fallen but the situation would not be good. Another option was that we'd have seen Iranian troops in Iraq which would cause problems within Iraq. Initially (2014 and 2015) it was the PMU and ISOF which took the upper hand in operations, 2016 and onwards it was ISOF, the army and police units that took the upper hand which is due to US-led coalition restructuring and retraining of these forces. It has been a very effective effort obviously so importance of US contribution cannot be denied here.
ISOF has always been a very effective force, this once again is due to its proper formation, governance and maintenance of the force which is a US effort initially. Iraq has been too disorganized to do this alone without foreign help and that help better comes from the US rather than any neighboring country as that would meet strong disapproval from locals and neighbors have plans that undermine Iraq whereas the US has different interests.