One major element in this is the rise of earnings from pass-through income. Self-employed or nearly self-employed individuals who would have been counted as employees of their small businesses 30+ years ago increasingly take "profits" from their business rather than "wages," even though they are the same thing. The "profits" get tax advantages. This makes labor share decline despite dentists, consultants, etc. still doing the same amount of work and taking home the same pretax income. See: https://www.ericzwick.com/labshare/labshare.pdf
Is it though? Self-employed individuals who've set up an umbrella business entity aren't particularly common.
By this paper's own conclusion, this phenomenon explains 31.9% of the decrease in recent decades. That's significant, but also means that most of the decrease comes from somewhere else.