Mind you, the default assumption would be that the money does just sit around in a trust, like with the child actors of today. The parents—having paying jobs, since that's what's required to be a parent now, and culture doesn't change that quickly—would be using their own money to support their child's upbringing (for status competition, if nothing else.)

It's only families that would currently be on welfare—and thus be auditable by that branch of government anyway—that would be spending their children's money to take care of those children. The effect of such a law would just be to take the auditors from the welfare and disability departments (which no longer exist under UBI) and move them to just making sure people aren't spending the money of people they have power-of-attorney over on themselves.

ETA: this logic all already exists in cases where children e.g. receive an inheritance from their grandparents. The parents are put "in charge" of the child's money, and are expected to only spend it on things the child would—if they were an adult—have decided to spend it on (like food to prevent going hungry, or schooling, etc.), and to otherwise leave the money alone. But this isn't enforced in any way, so this kind of inheritance is currently pretty much always squandered by parents.