The thing is to me, this aspect of both the clutch argument and the closer argument step all over each other.
In clutch arguments, the hitter is credited with responding to pressure. In closer arguments, the requirement is always stated as a pitcher who can handle the pressure.
There is pressure on both the hitter and the pitcher in high leverage situations. At some point, this cancels each other out, which is probably why most hitters and pitchers play to their career norms in these situations, and this becomes more noticeable as sample sizes grow...