Interestingly, over at Baseball Reference they list Keith Miller as the most comparable player to Gleason. I can remember Miller as a utility guy for those good Mets teams of the late 1980's. Miller was also one of the guys shipped over to Kansas City in the Saberhagen deal.
Jack Gleason...started out playing ball with the Stocks team of St. Louis in the early seventies. Later (he) joined the St. Louis Reds. (He) also played with the St. Louis Browns, the co-operative team of 1881. In the following year (he) joined the St. Louis Browns, the first member from this city to join the American Association...Later Jack Gleason went to Louisville...Jack Gleason was the hardest hitter St. Louis ever gave to the professional field. He was one of the few men able to drive the ball over the left field fence in old Sportsman's Park, a feat he accomplished early and often. He was also one of the best of third basemen.-The National Game
I believe that Gleason played for the Reds in 1876, after they dropped out of the NA. While no person played for every 19th century major league St. Louis club, Jack Gleason came the closest.