Simply not true. I was there. Many a Christian fled, by the thousands once the invasion happened. The Chaldean's in Mosul especially have taken it on the chin. It has been a Christian Winter for a while in Iraq; before BHO took office.
I was in Baghdad and Fallujah in 2003 and 2004. You are correct that many people fled from Mosul and other cities, but it seemed to be mostly in response to fear of what might happen. Many came back into Baghdad and and the immediate suburbs as we brought those different areas under control. Although there were attacks against Christians and other groups, those were sporadic and happened about the same frequency as other terrorist attacks.
After the surge, most attacks disappeared entirely, which allowed President Obama the political cover to withdraw US troops. This is when the real atrocities and attacks began in earnest. But I was only in one small part of Iraq, so I cannot make that blanket statement. But missionaries were operating in Baghdad, and The Word was being spread.