I think there are some valid points on both sides.
The fact that the books mention him collecting arrows off of dead bodies it a life saver for me! Other wise it would just be a urking inconsistency to me.
In the movies, however, it does not show nor make that as clear so the whole thing in the BOT5A to me is a little like... "Right.... here of all places he just HAPPENS to be out o arrows."
The timing is obviously suspense convenient. Personally I would have tried to figure out a different way for him to "not be able to save her." just because of the discussion (like this right here) I KNOW it will prompt.
However, it does stand to reason, that he would run out of arrows eventually.... so I'm betwixed and between.