That is not entirely correct either - it is more of a 50/50 joint venture between players and tournaments.
- Players elect 12 of their number to the Player Council, and tournament owners do the same for the 13-member Tournament Council
- The Player and Tournament Councils then elect 3 board members each (one representing the Americas, one for Europe and one for the rest of the world)
- The six board members then recruit a chairman from outside the organisation
The Player and Tournament Councils then serve in an advisory role to the board members they elected.
So the 7 member board's decisionmaking is theoretically split 50/50 between tournament and player interests, with a neutral chairman to break any vote ties.
In practice, the multiple levels of representation mean that your average player doesn't have a ton of influence over the ATP's decisions - so it makes sense that they want a player association to represent them more directly.