First, about children's ranks vs. adult's ranks...
Child black-belts are not uncommon in the world of Karate and Taekwondo. But when there are child black-belts, they are generally awarded that rank in the "children's" rankings. This rank is not generally the same as an adult black-belt. At least in most schools.
In some schools, there's no distinction between a child's black-belt and an adult's black-belt. They're the same rank. Though, this is not usual.
In Japanese karate, they will often use "mon" for child and "kyu" for adult colored belt ranks. For black-belt ranks, they award "dan-ho" for children and "dan" for adults. And usually children are only allowed to test for 1st dan-ho but not allowed to test for 2nd dan-ho, so they can't go any further than that until they reach a certain age.
In Korean karate (Taekwondo), they will use "gup" for colored belt ranks. Usually that rank is given for both children and adults. For black-belt, they'll use "pum" for child ranks and "dan" for adult ranks (typically aged 16 or older). In addition, child black belts will be half red and half black in color. Children are usually kept at 1st pum until they reach 16, when they can test for 2nd dan.
Now, this raises the question: Can children really be black-belts? What does "black belt" even mean?
In Karate and Taekwondo, usually rank is awarded based on: 1) Amount of time spent in the current rank, 2) forms which have to be learned for each rank, 3) self-defense techniques and "single-step" drills, 4) sparring, and 5) board breaking.
There may also be physical requirements, like being able to perform a certain number of push-ups, being able to run a certain distance in a certain time, etc. And students often have to memorize terminology and history and sometimes write essays. Most schools introduce sparring after white or yellow belt and board breaking only in the intermediate and advanced color belt ranks, but this is not universal.
So, if a child has indeed met all of the requirements for black-belt, there's no reason not to award that child a black-belt. That is what most schools do.
There's no requirement, for example, that a child be able to win in fights with adults. They just have to know their forms, remember their one-steps, know terminology, etc. Children can certainly do that.
Okay, so child black-belts are different from adult black-belts. We get it now. But does that mean that child black-belts can forget their forms, look sloppy, goof off during class, and generally look like they're no better than yellow belts?
Hell no! If you see that, it's time to look elsewhere. That student should never have been promoted to black-belt, even if it's just a "children's" black-belt.
Sadly, why you see this in many schools today has more to do with money than anything. Kids aren't going to want to stay in a school that's too hard or too serious. It's not fun, except for maybe a small percentage of kids that can handle it and enjoy the challenge.
And the parents aren't going to like it if the instructor isn't continuously promoting their kids. Parents want their kids to succeed. They'll pull the kids out of there and put them in a "better" school if their kids aren't advancing fast enough. It's actually pretty common for parents to get angry and confront instructors, demanding to know why they haven't allowed their kids to test for the next rank when all of the other kids have.
All of this translates to less students and therefore less income for the karate school. So instructors will lower their bar. That's why you see child black-belts in general. And it's especially why those child black-belts will look terrible.
Now, I've seen some pretty awesome kids who can perform karate forms and sparring much better than most adults. Those kids deserve their black belts. But the vast majority of children don't. And it's because their instructors lower the bar for them, usually to keep them as paying customers. They rely on children to make their money, because they usually outnumber the adults 5 to 1.
Now, here's the thing. If you're a kid, you should look at the other kids and see how they are before joining a school. If you're an adult, however, you should ignore the kids classes and just focus on the adult classes.
Often times, you'll see a different standard of skill levels and much higher levels of seriousness in the adult classes compared with the kids classes. It's so different that you might wonder how the kids could be trained by the same instructor as the adults are. And in fact, the kids classes are often taught by assistant instructors or advanced ranked adult students instead of the head instructor.
And definitely compare with other schools nearby. The instructor makes a world of difference.
So that's my advice. Hope that helps.