It is indeed a major betrayal of trust, if Els is using her security clearance and data access as a sort of quasi-Facebook to get juicy inside info on the MC's private interests and affairs. However, if she suspects that there is something more sinister happening, then her intentions could turn out to be less selfish than it might appear at first glance. My guess is that it's the latter, but we'll have to be patient and see what happens next.
Part of my wait-and-see attitude here is also a result of the MC's chill response to learning about the activity happening around his various records. IIRC, he just thanked Alina for the heads up, and asked her to let him know if she learned anything else. Nothing more. He didn't seem at all bothered by what people might find there, likely because he is already familiar with dealing with his negative public perception as a philanderer, so he isn't at all concerned with what people might think of him because of his military service or his brushes with law enforcement. As a result, that invasion of privacy means less to me than it probably would otherwise.
Els used her contacts within the british military intelligence community to have them access the MC's military records, in the hope of finding out what trauma he has suffered. The fact that it triggered an CIA response should be proof enough, that Els didn't go for any public data or files, but military, so she can't have had any altruistic concerns. She asked the MC, he didn't tell her, so she did it behind his back. Major betrayel of trust seems appropiate, no if's needed.
The reason the MC isn't concerned is because he knows that there is no way to get the information she is after that way, even without having a friend in the CIA. People would find either nothing or something that is entirely fabricated other than basic data, like the MC's birthday, which Els suddenly knows.
People, even friends, don't get to decide what Els knows or doesn't know, only she does. And I completely agree with what was said before, it's also the only reason why she interacts with Vi, she's just another source of information.