Somehow I had missed this earlier paper by John Charles Bradbury:
Since the early-2000s, the share of revenue going to Major League Baseball players has been diminishing similar to the decline of labor’s share of revenue observed in the US economy. This study examines potential explanations for the decline in baseball, which may result from related factors and provide information relevant to explaining this macroeconomic trend. The results indicate that the value-added from non-player inputs, collective bargaining agreement terms, and related changes in the returns to winning contributed to the decline of players’ share of income. Competition from substitute foreign labor and physical capital are not associated with the decline in labor’s share of income in baseball.
There is also this sentence:
The decline in labor’s revenue share in MLB is consistent with changes in revenue share in the hospitality and leisure industry that experienced a decrease in labor’s share of income from 65.7 percent to 62.1 percent between 1987 and 2011 (Elsby, Hobijn, and Şahin 2013).
Another hypothesis I have heard is that baseball players are not nearly as good at, or as well-suited for, the use of social media, as compared say to the more visible basketball players. Another (quite speculative) claim is that sabermetrics has commoditized a lot of players and in turn lowered their bargaining power.