“They paid for the seat. Doesn’t matter which kid used it. “
But it does. Who paid is not at all important. The name on the ticket is the only person entitled to sit in that seat. Period. If they didn’t change the name then they can’t use it. The ‘owner’ of the seat was not there, regardless of who’s money was spent.
This appears to be the crux. The claim was made that it was changed at the gate prior to the flight, and the gate agent was okay with it. That remains to be seen.
That being said, the father still bought all the tickets, it was still his family flying, there shouldn't have been any issue to begin with. As I said, I know in the past, this was never an issue, the ticket could be handed to someone else and they could board with no issues. Just check in at the gate and change the name, if even that. Post 9-11-01, it got really stupid, really fast.