BJJ is one of the most "useful" when it comes to actually fighting (along with boxing).
Karate and TKD are more of an art/discipline. A well trained karate fighter will very likely outmove an untrained assailant, but someone moderately trained in BJJ will likely be able to subdue/get away from a very well karate or TKD fighter.
Honestly, BJJ is an amazing skill to have in a pinch, and it trains you in grappling with opponents that have a size/weight disparity.
Not all gyms/dojos use belts, even in BJJ. BJJ belts follow a pretty good progression based on skill, whereas karate (can't say for TKD, never trained in it personally) often relies on performance of kata in order to progress to the next belt. Kata is choreographed movements, it's more like a dance that you practice than an actual measure of ability to spar/fight.
If OP wants to get their kid into a fighting sport that's fun and relatively safe, they can pick any discipline. If they want the added bonus of their kid being much better equipped to defend themself from a real aggressor they would do best getting them into BJJ, boxing, and then wrestling once they're in middle/high school.
I would personally avoid boxing for my own kids due to the repeated head trauma and risk of fractures, but it's the best real world striking training you'll get, at least in the USA. BJJ and wrestling help you immensely once you're on the ground, which is where 90% of street fights go within the first couple seconds, but a real, dangerous, fight is often over before it starts and countering a sucker punch or landing a decisive one yourself before the opponent can react is often the most important thing.
One of the downsides of BJJ is that it's culturally tied to MMA in the USA now, which means that if OPs kid does BJJ for a while in their youth they'll be more inclined to get into MMA in early adulthood, which is not something I would want for my children. But it's a great skill regardless.