That would seem to be the case for most rational people but time and time again we see millionaire athletes who don't seem to subscribe to this line of thinking.
I suspect this has more to do with the pressures other owners, other players, their agents, and the associated institutions bring to the public face of the sport; not to mention the general silence around contract and employment negotiations in the United States.
In a scenario where it is not all about the money, any self-interest scenario of the individual actors I can come up encourages both parties to keep it quiet and out of the public eye other than what sounds like the usual platitudes.
This seems much more likely than generalizing the near mythical greedy sports player mentality to most or all the players. Do I have a shred of proof? Nope yet it seems reasonable conjecture and more likely than assuming irrational greed across the board.