Haven't played it myself, but this seems related: https://slatestarcodex.com/2013/02/22/dungeons-and-discourse-third-edition-the-dialectic-continues/

Roll For Shoes is an exceedingly simple system. Like most, it relies on reading and writing skills, but I'm sure it could be adapted to work with simple drawings or even stickers.

To paraphrase:

• Each player starts with a single skill - Do Anything (1). The name indicates its scope and the number tells how many d6 to throw on a skill check using that skill. I think it works really well to use index cards or sticky notes for each skill.
• When a player rolls above an opposing check, they succeed. On failure they receive a pity point (some small token). In the context of playing with small children, it may sometimes be more appropriate if on failure they still do what they were trying, but not very well, resulting in some negative consequence.
• If a player rolls all 6s, they get a new skill based on what they just did with one more die than they were just using. e.g. Player succeeds at boiling water for the cook using Do Anything (1). They rolled a 6, so they learn Basic Cooking (2) as a new skill.
• After success has been determined, a player may use pity points to change dice to 6. This helps them level up their skills more easily.

That's the whole system, and the website has some settings that could be adapted to your kid's level of storytelling ability with a little creativity. I haven't been able to try this out with my almost-four-year-old yet since he hasn't been interested in cooperative storytelling so far, but I'm certain he could handle the mechanics provided we went to pictographic representations for the skills. You could even manage all the skill cards yourself if it's more fun for your kid that way.

Good luck with this, and I'm curious what else you come up with!

For story inspiration, check out AI Dungeon 2. It's basically GPT-2 trained on second-person text adventures.