clearly he has some pathological fear of the free throw line, and this explains it.
there are weird mental blocks in sports, like catchers that can't throw back to the pitcher, or chuck knoblauch suddenly not being able to make routine throws to first base.
not every buzzerbeater team is like the NBA, there are so many more players in buzzerbeater than the NBA you are going to see crazy things like this, but rather than insist that everything look "average" try to look at it with some creative glasses and humor, it will make the game much more enjoyable. Or from an analytical point of view, it seems to me that even a small amount of FT training would improve things dramatically for this player, and if you are doing a proper good cost/benefit analysis, you will train him, make more FTs, win more games, etc... which is i think the point of being skilled at the game.