It largely hearkens back to the role of "manager" evolving from the "captain" role. Captains were frequently player/managers like Frank Chance whose managing career began while they were playing regularly, and their eventual retirement came softly, when folks realized a guy hadn't put himself into the lineup in a few weeks. Maybe he was still available to pinch-hit, and maybe he did on occasion. But when he eventually didn't, by the start of the next season, he was understood to be retired, but as his role was only gradually changing, there was no moment that presented a particular impetus to change the clothes. Plus, he was still taking the field before games, to throw batting practice and hit fungoes and play Pepper and shit. The uniform was considered to be athletic wear, and there was no other generic athletic wear as there is now. Grey sweats were sort of a thing, but not really dignified outside of the gym. He was still sorta/kinda an athlete, and so the duds persisted. What's just as interesting as the uniform persisting as those factors grew less prominent is Connie Mack breaking with tradition all those years, and others not following him.