I wish the airline had NOT given the money back. Like a couple of PPs I do not think this is in any way the airline's fault. And WHY should the gate agent be in trouble? Their job is to sell the tickets, not interrogate the purchasers about why they are buying them or how old they are.
Geez, my 13 year old DD has been mistaken for my 11 year old son's mother on a half dozen occasions. An older looking teen ACTING motherly seems plenty old enough to do this.
I completely agree with both of the above posts
Oh and isn't possible that he kids DID have ID? Mine have passports and know where to find them.