There's nothing wrong with this per se except that it would be careless and stupid. It is a long established practice for anybody visiting Israel who is also a) planning on visiting any Arab countries using the same passport, and b) in possession of brains and/or common sense to have Israeli customs stamp a piece of paper for temporary inclusion in your passport instead of the permanent body of the passport itself.
Nobody working for the agency or even anybody with any knowledge or experience with the region would be unaware of the risks of entering Iraq with an Israeli stamp in his or her passport. Nobody.
I read on another thread Berk was at Cornell in engineering dept and on deans list for a couple of years. He was smart. He went abroad to Uganda in 98 for semester abroad in January and did not go back to Cornell, was on leave in the fall. At the same time in Uganda Al Quaeda was trying to set up cells to create violence and they failed. When Berg returned he was malnourished reported by his family to have given up his food to the natives. Now around that time the bombings occurred in Kenya and jTanzania if I remember correctly so Al Quaeda was pretty active in Africa. Am I reading to much into this or was he really malnourished from giving up his food or from something else. Then he never went back to Cornell. This man may have odd judgement but he is no dummy.
I know a Jewish guy who spent gulf war I at the Israeli pentagon, representing a US defense contractor. He had his own body guard, which is pretty cool. Anyway, he also did business in Kuwait and Saudi, and he confirms what you said.