Well, you may have nailed the problem. No, the trainees stamina is a 1. That probably explains it. I've only been paying attention to game shape, and his game shape is 9 at the moment.
Does game shape get better by itself over time? If so, how? Or does it only get better with stamina training?
Game shape improves or declines based on minutes played in a week, and it happens Friday morning along with your regular weekly training. If a player plays less than 48 minutes in a week, or well over 70, you're likely looking at a decrease in their game shape. If you get them in the range of 48-65 or more specifically 55-62 (in my experience) you're looking at an improvement. I wouldn't advise training stamina/game shape unless you're in really bad shape... managing minutes is more than enough, and it allows you to train actual skill attributes.
As for stamina, that regulates the consistency of play over the course of the game, so a player with low stamina will get more tired and become less effective down the stretch. Does your player seem to outperform other guards early and THEN fade, or is he just weak from beginning to end?