I don't have a problem per se, with coaches being paid for additional training (i.e. privates). If the goal is to help kids develop their soccer skills and the additional time is warranted, then more power to all parties concerned.
To the extent any coach makes decisions that impact playing time solely of extra dollars through privates, then that coach is ultimately shooting themselves in the foot and will find themselves in a constant battle of retaining players because those players that are unfairly discriminated against will eventually find other clubs/teams/coaches that are fair.
My son is a GK, and over the years his coaches (and other coaches) have asked him to join a private session or two that they have been giving to a player in order to support the private lesson (strikers taking shots), which he has been happy to do because it means free development for him. Those kids that take privates generally are the better players on the team and/or bubble players that are trying to make the jump up to starter.
In the one instance I have seen a coach (no longer in California) overtly favor his "private" lesson players, he struggled year in and year out to hold teams together because the parents of the good players tired of the Coach favoring certain weak players, thus, reducing the competitiveness of the team, and the parents of the bubble players left for greener grass, knowing this Coach wasn't interested in fielding a competitive team, rather the dollars.
So in sum, the problem exists, but Coaches quickly learn that who plays has to be based on merit, otherwise their teams fall apart (and their income drops).