i have been in search for the perfect RPG to play wth my kids (6, 6, 8 and 10). I've played a bunch of them—No Thank You Evil!, Tiny Dungeon: Hatchling Edition, Hero Kids, and now Amazing Tales. They all have pros and cons, and sometime I'll have to write up a comparison, but as much as I like NTYE and the others, Amazing Tales is now my go to for my two youngest. I have yet to try it with my two older kids, buy I think they will like it (though they are capable of understanding more rules than Amazing Tales requires.)
I'm pretty new to DMing, and this has helped get more confident improvising. Of all the games I mentioned, this one requires the least prep—not just in terms of character creation, but in feeling empowered to sit down without a plan and create a story together. It was very easy to understand, while also FINALLY providing a fun way for my kids to use polyhedral dice (all the others use either single d6 rolls, or various d6 dice pools. They work well, but I've been looking for a simple game to get my kids used to the polyhedrals.) Amazing Tales can be played with only d6s, but typically uses a d6, d8, d10 and d12, one for each "proficiency"
Highly recommended. While I still encourage any parent to get NTYE, and I think NTYE does a better job explaining how to play an RPG to a family who has never ever played one (and has fun game pieces to use), I'd recommend Amazing Tales over it for parents of 4-8 year olds in most situations, as a starter RPG and then trying any of these others if they liked it.
(for anyone who's intested, I'd say Tiny Dungeon feels like micro (and I mean really rules lite) D&D done theater of the mind (and you have to bring the story), and Hero Kids feels like an RPG with a board game feel, but very little prep needed as the adventures you can buy have lots of boxed text and every enemy and obstacle accounted for. NTYE has very open character generation, and rules to support that variety, and very fun kid friendly adventures). Amazing Tales feels like structured story time with the kiddos.