This thread has been a lot of speculation and not much data. QuestDog has said over and over that fatigue doesn't exist, others have agreed, or thrown in some subjective information.
So what really happens to players who play every single day?
Cal Ripken's OPS peaked in July (820), and tailed off to 766 and 753 in August and September.
Miguel Tejada's OPS was 838 in July, 821 in August, and 796 in September.
Steve Garvey peaked in September.
Retrosheet doesn't have Everett Scott's data.
Billy Williams had an .890-900 OPS in May-July, then was in the 830 range in August and September. He also played all of his home games in the day.
We only have the splits of the tail end of Stan Musial's career, after his consecutive games streak was over, but he played much better in June than in September.
Same for Eddie Yost, but his limited splits are all over the place.
Pete Rose was pretty consistent throughout the year, but his September OPS was 15 points off his July peak.
Dale Murphy was also very consistent, but his two lowest OPSes were August and September.
Does any of this prove anything? No, it could be that league OPSes peak in July and tail off through August and September due to weather or some other factor. But there does seem to be at least some evidence that players who are known for never taking a day off hit better in the middle than at the end of seasons.
Which may or may not be evidence that fatigue needs to be modeled the way it currently is.