The issue, from my standpoint, is a measure of understanding rather than physical skill. A 12-year-old has a very immature brain (clinically speaking) and is unlikely to have a depth of understanding I'd require of a black belt. Frankly, most 16-year-olds won't either, because most of their training would have come when their brain was very immature (it doesn't finish maturing until mid-20's for most people, as a frame of reference). For those reasons, I don't teach anyone under 16. If I did, it would be a different curriculum than the adult classes.
I'm not saying a 16-year-old can't be worthy of a black belt, even in my curriculum. It would just be rare enough that I don't feel the need to allow for it. If a student showed up who was capable, I'd train them to their level, and even consider making an exception on the promotion (one of the benefits of being outside an association). In mainline NGA (following the NGAA curriculum), it's technically possible for someone to get their black belt at 16. That's a stated minimum age. If they started training earlier, they'd start with the junior curriculum. After completing that, and when they are large enough/old enough/coordinated enough, they could move into the adult curriculum. If they completed the adult curriculum to the right level by 16, they could earn their black belt. To my knowledge, there haven't been any that young, but it's possible. I know one guy who got his at 17 or 18 (just before going to the Marines), but he'd been training since about age 8, and he was a fairly skilled and coordinated kid even when I first met him (he was probably 12).