That's what should have happened, yes, and I still hold out some hope that some of this was done. There are several barriers to that, though. First, they probably didn't have the trained manpower to tackle what would have been a huge and delicate undertaking. Second, they probably would have stored the results on-site, in which case they may be lost, too. They didn't want copies of these texts floating around, because they wanted to reserve the right at first translation and publication.
Unfortunately, they had made very little progress towards that end, because translating Sumerian is an extraordinarily difficult task. One reason is because there was no Sumerian dictionary until very recently. At the end of 2002, the University of Pennsylvania announced the completion of the first such dictionary.
A petty and fatal decision, no? Entirely the Iraqis' and far beyond the reasonably expected ken and reach of the military planners.
Imagine archeologists being criticized for overlooking a detail in the complex plan for an enormous military campaign!
It also borders on an asinine postulate.
The Dead Sea Scrolls are arguably no more important (in the larger scheme of things) and much more recent, and yet , in that instance, among the first tasks undertaken was the recording and storing in several diferent places throughout the world the raw material before any translation was undertaken.