As for COIN and timing, you will note that the prerequisite is defeat of major force concentrations, and that implies control of turf. The situation was not comparable to “withdrawn into more remote parts of the country” after defeat.

The primary sin of omission was doing obvious law enforcement in the face of overt abuse and looting, etc. The lack of personnel to do much more was the result of the two-fisted grabbing of “peace dividends” from defense spending by Clinton, and even today Congress won’t facilitate unwinding of that inanity.

As Crocker notes elsewhere, there were two generations of disenfranchised and civicly ignorant Iraqis making up the bulk of the population. Their education about who was/is the “strong horse” probably had to come the hard way. There is an interesting sidelight: the younger of those two generations is resentful and contemptuous towards Islam, as it was, from their POV, the club used to beat them down half their (short) lives, and latterly condoned by silence and demagoguery the indiscriminate murders of their friends.