The reason why is probably because he played most of his career with the Phillies, who were really, really terrible. He did play on a pennant winning Cub team in 1935.
In that year, 1932, he also led the league in runs, hits, slugging and total bases.
In that year he won the MVP (he was second in the voting the year before and the year after), the Phillies were two games over .500. Not only was it the only year when he was on the team that the finished over .500, that 1932 season was the only year they finished above .500 from 1918 to 1948. (Yeah, they're pretty doggone bad now, too, I know). Most years Klein played for them, they lost over 90 games.
On that single season outfield assist record, he had 44 in 1930. I'd add that to the "records that will never be broken" list.