Did the achievement glitch on your account? No, all achievements unlocked exactly when they were supposed to.
Did you start out on SuperSoldier Difficulty? No, I beat it first on Easy then on SuperSoldier
Did the Veteran 3/7 achievement unlock? Yes.
Did you access the tutorial from a new game or through the menu? New Game. Started a game and accessed Tutorial when it asked me.
Prior to campaign completion, did you replay missions through mission select? I did not. I started the game on Easy, beat it. Started on SuperSoldier, beat it. Did not select missions until after.
If it didn't unlock initially, how did you finally get it to pop?
Did you press the X button to skip cutscenes either at the end of a chapter or during a mission briefing? I skipped every cutscene I could, when I could.