That kid did an amazing job, BUT human lives are in play even when it comes to takeoff clearance, so yes this was unacceptable.
There's a difference between giving a blind takeoff clearance and a supervised takeoff clearance. If the controller had left the youngster to give takeoff clearances and departure handoffs to himself, then yea, that would be an inexcusable problem. But no, the controller was (and presumably an additional supervisor, and heck, probably more certified controllers were) undoubtedly watching his movement areas at all times when the kid was issuing instructions; he was also issuing landing clearances on the same frequency. This controller (you can tell by his voice) has been on the position forever and knows what he's doing. Logically speaking, he wouldn't risk his job by performing an unsafe operation. Radio instructions are only a fraction of what it takes to issue a safe takeoff clearance. Scanning the runway, coordinating with the departure controller, scanning the arrival corridor, knowing who's up next, knowing the wind, looking at the radar feed, and so much more is involved with the issuance of a takeoff clearance than a small child could handle, which is my reasoning for supporting the controller. He wasn't being stupid about it.
There's no reason to believe that safety was compromised, however, I understand that is the reasoning behind the investigation.