In his homily on John 16, St. John Chrysostom explains John 16:22 in this way. Notice the elegant way in which he ties the sorrow and joy of the disciples to Jesus life, death and resurrection.
And yet the woman does not rejoice because “a man has come into the world,” but because a son has been born to her; since, had this been the case, nothing would have hindered the barren from rejoicing over another who bears. Why then spoke He thus? Because He introduced this example for this purpose only, to show that sorrow is for a season, but joy lasting: and to show that (death) is a translation unto life; and to show the great profit of their pangs. He said not, “a child has been born,” but, “A man.” For to my mind He here alludes to His own Resurrection, and that He should be born not unto that death which bare the birth-pang, but unto the Kingdom. Therefore He said not, “a child has been born unto her,” but, “A man has been born into the world.”
If you are at all familiar with John in the original Greek, after reading Chrysostom’s comment, this meaning practically jumps out at you. “A woman has sorrow, but that sorrow is turned to joy. How much more then will our sorrow be turned to joy, seeing that ‘a man has come into the world’ to save us from sin death and the devil.” (And no matter how hard you try, you simply can’t un-see it)This sort of wordplay is a very Johannine thing to do. So, how does the ESV handle it?
When a woman is giving birth, she has sorrow because her hour has come, but when she has delivered the baby, she no longer remembers the anguish, for joy that a human being has been born into the world.
Oh, well then… never mind, I guess.