There are a few RPGs out there specifically designed to be kid friendly, although what they interpret those words to mean can vary a lot from one product to the next. With my limited sampling of one, I have noticed that --for my household, at least-- there are some minimum expectations I (and my son, and wife) have for what a good kid-friendly RPG should look and behave like.
First, and this is the sad part: it should look good. Like, really good. It doesn't need to be wild and crazy full color art (though if you can manage it, please do) but it needs to be evocative and interesting. It should help spark the imagination of the kids, who in many cases have arrived at tabletop gaming after plowing through video games, tablets and other venues.
Right now, two of the games my son is most interested in are Pathfinder and Starfinder. The reasons should be obvious: Paizo knows how to make a good looking game, one with iconic depictions of the kinds of characters you can meet or play and the kinds of monsters you can fight. Everything in the game, to a greater or less extent, has an illustration accompanying it that just begs for the PC, NPC or monster (or starship) to jump off the page and join the story.
There are some kid-focused games that do the art well. Monte Cook's No Thank You, Evil! is a good game written specifically for kids that is full of great, evocative illustrations and lots of parts and pieces. It's main issues is one of thematic content and it's actual intended audience, about which I will discuss in a moment, but the game fits the bill here. It is also written at a sort of "parent level" for most of the text. Older kids will get it, but for younger kids there's no supplemental booklet I am aware of that you could hand them right now to help learn the game without parental guidance.
Unfortunately, and this is the second point: Paizo writes games for older teens, college kids and full on grumpy old adults. Their books are not written to be introductory, and in fairness not even the Beginner Box for Pathfinder is a good introductory book for kids, although it makes admirable steps in that direction. That said, some games are written well enough for a nine or ten year old to pick it up: Tiny Dungeons has a version intended for this purpose (though the core rules are accessible to a kid of 9 or 10 just fine). Tiny Dungeons has some cartoony, somehwhat evocative artwork to go with it, but pales in comparison to its big budget adult competition (however I'll note that Tiny Frontiers: Mecha vs. Kaiju solves this problem with some awesome art, fyi).
Now, when I think of "kid gamer friendly" I am thinking of rulesets that are written for kids, and intended to teach the kids without requiring any more than minimum adult intervention. In my day, at age 10, I was able to figure out Gamma World 1st edition on my own, but only after spending months trying to parse out the Otus cover D&D Basic book, while begging my dad to decipher it for me. In the end, for some reason Gamma World spoke to me in a way D&D Basic was missing, and my first RPG game ever was a Gamma World scenario as a result.
Neither of those books were terribly kid friendly on a certain level; but kid friendly doesn't mean "dumbed down" so much as "accessible to read and figure out." In fact, if my own life experience is any measure, a certain amount of esotericism (the Gygax effect, if you will) in the text is useful to engage the young reader; it's why Harry Potter books are so damned successful, for example. They challenge the kid, and also offer him new and strange concepts that he can feel good about figuring out.
I'm not sure many games out there do this well right now. If there are any, I haven't quite found them, although I will label Tiny Dungeons and its lot in the short stack of games that I think are on the right track. Lone Wolf could fit this bill as well. D&D 5E, believe it or not, is definitely more accessible in this regard as well.
Oddly, I don't think OSR does this well. Most OSR games, while simple in design (and providing exactly the right level of complexity for what my son could learn) are written by old men (also called "dads" or "granddads") writing for other old men. Very few are written with a kid in mind.*
Likewise, a game like No Thank You, Evil! is not so accessible. It's actually targeting adults who want to game with their kids in a carefully sculpted environment, while overlooking what the kid really wants or needs.** It takes great pains to focus on a game experience that an adult (dare I say, helicopter parent) might want to curate for their kids rather than, perhaps, the kind of game the kid really wants.
I guess what I am trying to say is this: if you as a parent want to play a game with your child in which they solve problems with their magic hot-wheel trike while making friends of enemies and exploring a Candycane universe, then No Thank You, Evil! has that sort of sanitized child fiction setting down pat. It alludes in the rules to the idea you could do more....but refrains from actually suggesting anything.
But, if you as a gamer parent want to let your kid kill a giant goat and skin it, then save their friend from murderous bugbears by hosing them with fire, then D&D is kind of your best bet.
Put another way: I want my child to have experiences which challenge him with interesting but realistic decisions, and allow for the game to grow in complexity and meaning as he grows. D&D can do that. No Thank You, Evil! can't (well, it can --a bit-- but not in the sense I mean). At least, not in the broader sense that I want it to. Would my kid have fun playing NTYE!? Yes, he has and would. But he's going to want to have Punk Rock Demon blow up candyland if it doesn't play nice, and if we're going that direction, why not play the game where you can actually do that?***
I think Tiny Dungeons could do this, too....but ironically I suspect the rules would eventually fall behind the desired complexity over time. I mean....I've seen the games my son's generation loves. Minecraft only looks simple. It is, in fact, a remarkably weird and complex game of crafting, and my son is already pushing D&D to see what he can craft (e.g. goat meat).
My notion here is that there is a market for a game which accomplishes the following, all in one package:
1. Provides a graphically engaging and evocative portrayal of its shared universe in the art
2. Is written or structured to provide a progression over time in learning the rules and method of play (think Basic vs. Expert)
3. Is written with a kid in mind, rather than an adult, and assumes the kid is smart and can figure things out, or really wants to
....there may be games out there I don't know about that do this. I would welcome suggestions! But that said, I think my son will greatly enjoy D&D going forward, and I may adapt Starfinder content to the D&D rules, or perhaps reskin content for White Star, so he can enjoy the graphic universe depicted in the one game with a ruleset that will be explainable to him by dad (who frankly has enough trouble remembering all of Starfinder's rules without one of my rules lawyers at the table to assist!)
*Here's an example of what I mean: Swords & Wizardry Complete has some good source books with evocative art (3rd edition reprint is nice, although I prefer the 2nd edition look ultimately as an old grognard of sorts). Monstrosities and Tome of Horrors Complete both provide an illustration for every monster by decent artists, for example. However, try reading the S&W Complete book. A version aimed at kids would not need all that exposition on what the Founding Father Gary and Dave intended with initiative, to give one example, or the exposition behind intent of class limits or multiclassing. That's valuable space that you could place working examples of play or add additional useful content to game with. A good take on this is a ruleset that is instructive and provides plenty of exciting examples, but does not cut content; I'd argue that Beyond the Wall is a game that moves in this direction, though it is still written for adults and not kids....ironic, given it provides some of the best tools yet for aiding a new or young gamer in playing. Moving away from the "historical reference" that some OSR games provide, as well as the "OGL reskinned for OSR" format of others would help a great deal in accomplishing this sort of goal.
**Your mileage may vary, a lot. I could see NTYE! working well for some kids. Others? Not so much.
***I'm showing a little bias here. I just feel like NTYE! is the sort of game written for parents who are aggressively trying to control the sort of content their kid experiences. Any parent should do that, but there's a difference between "You're too young for this stuff," and "I am shielding you from basic life experiences and complex decision making scenarios." I feel like maybe NTYE! contributes more to the latter than the former, by design, since it is aimed at the sorts of parents who maybe worry that Little Johnny shouldn't be fireballing bugbears. The same sort of parents who won't allow their kid to play Fortnite or Call of Duty, maybe, but Candy Crush --a downright evil game by addiction design--is somehow okay. But I could be wrong.