Many relief pitchers pitch only from the stretch position because they have to make the most crucial pitches in a game with men on base. Most times a reliever comes into a game, it just happens to be with men on base in the pivotal win/lose point in a baseball game. Many studies have proven that close to 70% of pitches thrown in the game of baseball are with men on base. Hence the need to have your best stuff from the stretch. Relievers don't have the luxury of finding their rhythm during a game as starters do. They have to be spot on when they enter the game. Pitching solely from the stretch helps to stay more consistent night in and night out. Many amateur pitchers constantly work their bullpen sessions from the windup only because that is where they feel the most comfortable or that is the position they throw their fastball with the greatest velocity. Most Professional pitchers start from the stretch first when they get on the mound. This is done for many reasons. One, it helps to get their timing down before anything else. Two, they know they will need quality pitches at the most crucial point in the game from the stretch position with men on base.
Working your Side sessions (bullpens) primarily from the stretch will help build confidence to make quality pitches when you need them the most!