To answer OP's narrow question, there is no requirement that a US carrier seat a child adjacent to a parent.

However, DL will, if necessary, involuntarily move another passenger in order to assure that a 7 YOA is adjacent to a parent. This does not mean that the family as a whole is seated together, it could mean that the seats are across an aisle, and it also means that the adult is moved to one of the seats at the rear of the aircraft which are held for oddball situations.

If it were my 7 YOA, I would pay the money to assign seats.

DL goes out of its way to warn passengers about the perils of BE. No need to overcomplicate things. There is only so far a business can go in telling a parent how much she ought to care about her kid.