ISIL exists because Iraq is a Failed State
The present Iraqi government exists to protect and advance the interests of Shiite Muslims who are the majority in the southern part of Iraq.
The area in Northern Iraq presently claimed as part of the ISIL "state" is militarily dominated not by jihadist fighters, but by Sunni militias led by former officers of the Hussein regime.
The Sunni tribes who make up much of the population of Northern Iraq have legitimate and very serious grievances against the Shiite government. They appear to prefer the cruel, barbarous iconoclasm of ISIL (which is not directed at them) to systematic oppression by the Shiite government.
It appears very questionable whether a Shiite dominated central government in Iraq can ever gain effective control of the northern areas presently claimed by ISIL. "Iraq" has effectively been partitioned into the Sunni North, part of Kurdistan and the Shiite South.
Who cares about the Mosul Museum? The sad answer is that neither the Shiite government of Iraq, nor the Sunni tribes of Northern Iraq, nor the Kurdish peshmerga (the only effective opposition to ISIL within Iraq) care very much. They have much larger (and more important) problems to deal with than destruction of Assyrian antiquities and the cultural heritage of Assyrian Christians.
It isn't fair or accurate to blame the barbarous rampages of publicity-hungry ISIL jihadists on Shiite or Sunni Iraqis or the Kurds. Iraq has never really been a unified nation, and only overwhelming force (such as that used by Saddam Hussein) has ever succeeded in suppressing its ethnic and religious divisions and ensuing conflicts. The only way to stop these conflicts would be to accept the reality that Iraq has effectively been partitioned, and that Iraqi Kurds, Iraqi Sunnis and Iraqi Shiites have established separate states. That however is not politically correct.
Important foreign interests, including those of the United States and Saudi Arabia, would be adversely impacted by acceptance of a Shiite Iraqi rump state which would effectively aggrandize Iran. Turkey and Iran would oppose emergence of a Kurdish ethnic state. The realities of Mideast power politics will almost certainly prevent a stable political solution from developing until long after the ISIL crisis comes to an end. Meanwhile, the only factor capable of exerting any constructive influence toward curbing the jihadist barbarities of ISIL will be the moral force of religious authorities.
That is why it is essential that every Islamic religious leader should denounce ISIL as a destructive, evil perversion of Islam that all true Muslims should abhor.