In her 11 August response, Barr questioned whether the special agent who conducted the investigation “can be an impartial evaluator of academic scientists, or anyone with liberal political beliefs.” As evidence, she points to a posting on a blog maintained by the agent, a veteran who served in Iraq, and his family. The item is a copy of a popular Internet meme about an incident that supposedly took place in an introductory college biology course. According to the story, a “typical liberal college professor and avowed atheist” declares his intent to prove that there is no God by giving the creator 15 minutes to strike him from the podium. A few minutes before the deadline, a Marine “just released from active duty and newly registered” walks up to the professor and knocks him out with one punch. When the professor recovers and asks for an explanation, the Marine replies, “God was busy. He sent me.”
This makes it look really like this was a single agent who was unhappy with the left-wing views she had. At minimum, it is wildly inappropriate for a government agent in such a position to have that sort of thing on their blog (aside from it being just stupid). That goes together with the statement in the article:
Attorney Joseph Kaplan, of the Washington, D.C., firm Passman & Kaplan, says that, in his experience, the most common reasons for a finding of unsuitability are lying about one’s educational background, one’s employment history, or one’s criminal record. “If OPM determines that the person has misled or provided false information,” he says, “they can be declared unfit for federal service.”
Kaplan says he’s never heard of anyone being drummed out for political activity that occurred decades ago. At the same time, he says, the government’s decision is based not on anything Barr did during the 1980s but on how she explained those activities to federal investigators after coming to work at NSF.
Together this paints a potential picture of a specific agent going after someone they didn't like due to their political views and a bureaucracy going into overdrive to protect that decision. On the other hand, it isn't like she had no connections to the third organization- she knew two of the people who were convicted of the murder and by her own description kept up a correspondence with one of them while he was in prison. But keeping up correspondence with someone in prison is not evidence by itself of any problem, and there's really been no evidence presented that she lied or attempted to mislead in any way.
The article notes that this may be due to more general post-Snowded reactions which are making these sorts of things more common. In that case, this is exactly the wrong response.