While I do agree with you that this might have been a better look in some senses ("let's hear each other out privately as reasonable men"), I think you're overlooking the power dynamics that are (still) at play.
To take an extreme example (that is clearly not exactly parallel, but I hope gives a sense of something of the possible issues at stake), imagine that the charge was that a boss (or former boss) raped an employee. If the boss was confronted by a reporter about the accusation and then said something like, "I don't agree, but I'm going to reach out to the accuser and try to make things right," there would be all kinds of red flags raised about whether the boss was using his power to try to minimize the issue or even use the power differential he still possessed to make the accusation go away or even punish or otherwise regain power over the accuser in some way.
Now racial power dynamics don't play out exactly the same as sexual power dynamics, but I think there are enough parallels in this instance that Lindsey wouldn't necessarily have been better served to have gone the "let me try to make things right with Elijah privately" route. (In fact, I just saw a tweet from Millsap subsequent to his initial statement about the exit interview to the effect that he's less interested in "making things right" than speaking the truth with integrity.)
Whether Millsap intended things to play out like they have (and he might have, though sometimes once actions are taken, they lead in not-fully-anticipated directions), I think given the way the social climate and news media work these days, once the accusation went public it would have been hard for Lindsey and Millsap to just reach out to each other as more or less "equals." Maybe somewhere down the line, they might be able to do so, but probably not at first.
