In the case of the woman in the article who is a citizen of the US and who is applying for Romanian citizenship to obtain an EU passport, I would recommend that she use her US passport to travel between the US and Romania and then her Romanian passport if necessary for *work-related* travel throughout the EU. If she isn't working over there, then I would recommend that she use her US passport exclusively throughout the EU; once she enters Romania (or any other EU country) on the US passport she should have no problems traveling from one EU country to the other.
No, I retired at the beginning of 2000. I agree with your recommendations to trhe Romanian woman. The State Department recommends that the traveller should use their US passport to leave and enter the US.