The point is that the traveler disclosed to UA the child’s age (by virtue of giving the dob) and UA decided, after that disclosure, to sell the ticket. So UA technically waived its terms and conditions and therefore should not be falling back on them. It’s akin, in a way, to a mistake fare in that UA didn’t mean to sell the ticket but did so mistakenly. You can argue whether the traveler should have, or might have, known. But the principal blame lies with UA and it should be UA’s responsibility to sort it out at no, or reasonable, cost to the OP.