At what age can delta seat a child by them selves?
 

They should move him close to the parents. Ideally, next to one of them, the other parent seated with the other two kids.
They would charge a fee if no parent was traveling with a 7 year old, because by their own rules she needs care that can't be expected from a stranger sitting next to him by chance. They wouldn't let her travel unaccompanied by an adult, so they charge the fee to accompany the minor child.
Finally, my main problem with seperate seating like this is in an emergency, the parent is likely to impede other passengers as he wants to get his kid to safety - you know, be a parent. FAs will nag passengers to no end if p their baggage doesn't fit under the seat because it might impede other passengers in case of an emergency. This is far more serious.

I was curious and looked through Delta's website to see if they post details and ages that would help answer this question. The most I found was Delta's cheery: "We're here to help families take flight."

BUT.... they do go on to say that families should "... know the details about flying with infants and children."

Delta clearly posts that Basic Economy DOES NOT come with pre-assigned seats.

During the ticketing process for basic economy I really wish Delta would REQUIRE mandatory seat selection for minors to a certain age next to an adult in the ticketed party. Yes, RIGHT AT BOOKING FOR BASIC ECONOMY adjacent seat assignments should be MANDATORY when there are minor children below an acceptable age traveling.

And those seat assignments should carry a fee equal to the difference between basic economy and main cabin.

Basic economy fares are discounted because the ability to select seat assignments is not included in the discounted price. Delta is clear about this, so I'm not sure why parents are not.

Or maybe Delta should prohibit the sale of Basic Economy tickets to minors, so confused parents aren't disappointed.

Typically 15 years and younger flying without an adult must pay for UMC. Logic says a 12 year old could sit widely seperated from their parent without involving their seatmates. But the emergency issue is still out there frustrating the airline's attempt to monetize parenthood.
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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.

The issue is the question, if not purposely or knowingly done so, creates the controversy because a family bought a type of ticket knowing it didn't come with advance seat assignments is now in a position of possibly having to request a seat swap with a passenger who did not buy a Basic Economy ticket so they could select a seat in advance. Chances of getting middle seats spread throughout the cabin is also high when on BE so the chances thus become high that said passenger who is being asked to swap is being asked to swap from a aisle or window to a middle seat.

Edit: sounds like this is exactly what happened and why people were making remarks like "get the popcorn"...….

There's an easy way to do this - block out the last two rows of the plane for families in BE.

(and decrease pitch to 28" while they're at it)

WNK,BNK