Because counterinsurgency may look like war at the tactical level but it is not at the strategic level. To the extent we elect to treat it like a variant of war, we fail.
I think I would agree with you. The phrase "to the extent we elect to treat it like a variant of war, we fail" is apt. The "extent" is the issue. Obviously, counterinsurgency will contain elements of combat. The problem seems to be whether adherents of Clausewitz choose to jettison his commentary on political objectives and how much effect politics have on war. Or, in this case, counterinsurgency.

Of course, Clausewitz spent limited time on this part of his "theory" because he was ultimately a soldier whose theater was war. It didn't make his commentary on the political aspects less astute or applicable. It does mean that the emphasis should be on this extract of his thesis and supplemented with Mao and others who focus on the balance of the political with combat.

Not because the political aspect is the only "means" to the end of an insurgency. Particularly when that insurgency is extensive and very violent. However, politics are the cause of insurgencies, thus, politics are the final solution. In between combat and politics ebb and flow.

Soldiers are generally and by nature warriors focused on combat as was Clausewitz. They are conditioned for traditional war, to confronting the enemy with violence. Thus, the extent to which politics are emphasized in the education of military leaders and the common soldier is simply an attempt to balance out natural inclinations and previous training in order effectively conduct a counterinsurgency.

Because it is a counterinsurgency against violent insurgents and these counterinsurgencies, for the US, are expeditionary counterinsurgencies in other nations, it requires combat capable forces that can also act as political arbitrators. They must be able make war and negotiate peace at the same time.

Counterinsurgency is the balance of politics and war; the application of law and directed violence. It's yin and yang, feng shui, whatever term of balancing we want to use. It's just that the last weight that is dropped on the scale is always political.