Did the achievement glitch on your account?
Did you start out on SuperSoldier Difficulty?
No, my first playthrough was on normal with all collectibles + misc. achievements.
Did the Veteran 3/7 achievement unlock?
Yes, both Veteran and SS 3/7 achievements unlocked when they were supposed to.
Did you access the tutorial from a new game or through the menu?
Prior to campaign completion, did you replay missions through mission select?
Yes, I replayed level 3 to finish off the suit modules. I replayed it on the same difficulty of my first playthrough. Did not want to take a risk of something glitching because of selecting another difficulty.
If it didn't unlock initially, how did you finally get it to pop?
Did not glitched.
Did you press the X button to skip cutscenes either at the end of a chapter or during a mission briefing?
Did not skip any cutscenes on my first playthrough, but on my second (SS difficulty) did skip everything.