I have not seen "A Winning Team, but I read John C. Skipper's excellent book a few months ago on Alexander entitled "Wicked Curve: The Life and Troubled Times of Grover Cleveland Alexander." The book contains several references to the movie and talks about how the movie sanitized Alexander's real life struggles.
http://www.amazon.com/Wicked-Curve-T...over+alexander
In reality, Alexander was deeply affected by shell-shock following his combat service in W.W.I, had to have an ear removed due to cancer, suffered from epilepsy and was completely overtaken by his alcoholism once he was out of baseball.
Skipper's book mentions several times that most baseball executives and team owners viewed Alexander as an embarrassment to the game because he could not control his alcoholism and was seemingly beyond help despite many efforts to provide him employment or just outright financial handouts late in his life.