I think you can definitely read childrens’ books! I finished genki 1 and am only level 9, and I get by just fine with picture books. The thing that makes meaningful input effective is understanding the general idea of a passage. For instance, this book I found in Japan:
ミッフｨー is the main character. Genki supplies you with all the vocab need to read the passage, and you can infer the meaning of the other words. I think you should just read whatever you can get your hands on and not worry if you’re “ready”! Good luck!