You are right about the mental pic of the court. When I am approaching a ball and have my seeing focused intesnsely on the ball, and also have a '3rd eye' vision in my head that knows exactly where the baseline, sidelines, and corners are so I know how I want to swing and where I want to put the ball without looking up. This results in quality contact in the center of the racquet face.
Same for me with golf. Although my eyes are on the ball, I have a '3rd eye' picture of my target, although obviously a fairway or green is not as symmetrical and geometric as a tennis court, and you don't have the reinforcement of seeing the court lines and net out of your peripheral vision to aid in your aim. I have tried looking at the hole while making my actual putting stroke and it doesn't work for me. I can have a picture of where the hole is while looking at the ball, but I can't seem to picture where the ball is while looking at the hole. I know it is right in front of my putter, but I find myself relying solely on body mechanics to return the putter head to that exact same spot on the forward swing, and when I rely on body mechanics things get ugly...
Even with my dependence on sight and ball focus, I easily have the longest most complete follow-through of anyone I play with regularly, and I hit through the ball rather than at it, whether it's putting or the full swing. (Drew - when you take practice swings before each shot, focus on making full and complete follow-throughs, this may offset some of your tendency to hit at the ball)