The checker (for those who have automatic entitlement, not Transitional Protection, that's a census thing that has to be manually entered by the school) consists of entering the NI number, name and date of birth of the parent who receives benefit, the school name and the child's. If there's no child attached to a claim, there's no eligibility. Child benefit is paid on the basis of who the kid lives with - and if he lived with his father whilst she was in rehab, he'd be entitled to receive it.
The checker also works for NASS applicants, whilst kids entering the country with No Recourse to Public funds have to follow a school based manual process. It's potentially possible that during Covid, the council just added every kid in a household that was eligible for Council Tax Benefit and informed the school, but my point is that he'd only potentially be eligible through living with his father as his mother's income would have been too high, especially with a partner earning as much as LJC did.