Part of me can totally see Reid being the kid of JJ and Hotch, but it's pretty much canon that Rossi and Hotch are the "team parents" since Morgan said it in season 3 when they were gone
I feel like you're right that JJ usually deals with the kids that are involved in their cases. It's probably mostly her and Hotch probably does some as well. But I would love to see Reid be the one to deal with the kids sometimes. He's shown that he's grown and is great with kids. There no longer is a "Reid effect"