Unless I'm misreading something, Xarlen's players never referred to him as 'childish'...quite the opposite, overall. They said they were reminded of his age in this instance, "despite his maturity", I believe was the quote in the first post. No one accused him of being a bad DM, no one berated him directly, but there was obviously a lack of enjoyment on some player's parts. I haven't read the transcript, and I don't know the people involved, so I can't say for sure if Xarlen made a mistake or not.
I would put forth two thoughts. First, age CAN make a difference. Not necessarily from a maturity standpoint, but from a perspective standpoint. Xarlen is under twenty, some of his players over thirty. I can't speak for them, but I can speak for myself and those I know that our view of life and some topics changed dramatically over those years. My view of the acceptability of certain topics has changed. Other life factors can affect this, too. Once you've had children, you view child endangerment and abuse MUCH, MUCH differently than before. Life experience changes perspective.
From that perspective, I think Xarlen screwed up...NOT because of his age per se, but of his lack of appreciation of this fact. He didn't check with his players to make sure that he was going to cover some material that was potentiall upsetting to some players. It's a minor mistake, and it grants experience. If Xarlen is still upset with it, contact the player, make sure they understand that it was intended to upset anyone, and move on. Mistakes happen; no harm, no foul.
Second: I don't think this would have happened in a face-to-face, 'around the table' game. That's not a condemantion of online gaming, which I think is great (though I don't have much call to actually do it). Merely that Xarlen is gaming with folks he doesn't really know that well in the RW, I'd hazard, and so he isn't as familiar with their personal situations. Moreover, an online game doesn't allow you to read the reactions, attitudes and body language of your fellow players, to realize things aren't playing out as well as hoped. If he could see three of his four players not reacting well to the plot, he might have toned down the 'my children, my baaabiessss!' element, to the enjoyment of all.
In short, don't sweat it too much, Xarlen. Chalk it up to a learning experience that makes you a better DM, and move on. I would, however, try and make the player in question understand your motivations. I think that would be a benefit to both of you. Her personal issues are not your responsibility, true, but as DM, you have something of an obligation, IMHO.