I concur that a broad approach is needed; however approaching the problem from the default positions of COIN or CT is too limiting and may (if it has not already) stifle creative strategic thought. I think we are making a mistake in trying to put everything into a COIN or CT "box."
That said ISIL clearly has evolved from an Islamist extremist based terrorist organization. I know we do not want to legitimize it by calling it the Islamic State but I think we have to look at it as more than a terrorist organization. Although there are tremendous political problems in Iraq and those political problems have created security vacuums that ISIL has been able to exploit I do not think ISIL is conducting an insurgency against the Iraqi government. What ISIL has done has become an occupying power in northern Iraq and is not seeking traditional forms of legitimacy from among the relevant population. It is oppressing (and ethnic cleansing) the people living in its occupied territory and it is obviously using the tactics of terrorism to oppress (and ethnic cleanse) the population. It may have the objective to defeat all of Iraq and occupy it but I do not think that it will attempt to do so through insurgency and terrorism (though it will continue to conduct terrorism in support of its objective). It will conduct large scale military type operations to achieve its objectives and to ISIL the population is secondary and is either used for exploitation or is simply in the way and an obstacle to be eliminated (again ethnic cleansing). If we focus exclusively on CT and COIN we may very well miss the bigger picture and thus provide the wrong response to the wrong problem.
I think the broad approach needed is to not approach this from an insurgency or terrorism perspective but to approach it from an occupying power perspective. This might lead to a two pronged approach - one is conventional forces (perhaps Iraqi and Kurdish) conducting conventional operations to retake lost terrain and population centers and defeat the ISIL forces hopefully with the support of a coalition with air power, ISR, and advisory support. The second lesser and supporting approach might be an unconventional approach that leads to the overthrow of an occupying power if there is any remaining indigenous
resistance potential in the occupied territory. While coalition forces could provide external support to such resistance on the ground support might best be provided by non-western regional elements (from the Iraqi or Kurdish militaries), again with coalition external support.
But the bottom line for me is that using a COIN or CT approach is too strategically limiting and we need to address the ISIL threat as it really is (an occupying power) and not as we would wish it to be (an insurgent or terrorist organization).