I'd like to hope that at least for the majority of children who age out of the Imagination Library, both the child and the parents have "become people who read" and most likely the child gets a membership to their local library.

I remember my family getting taken to our local library every couple of weeks to return and then borrow books while I was in infants school. That made me 9 or 10 years old at the very oldest, and my youngest sister would have been at most 5 or 6 years old. So I'm pretty sure aging out of the Imagination Library is a pretty reasonable time to start that child using a "real" library, and start having several books a month available for free that way.

(I fully acknowledge the immense privilege I had having parents who understood and supported the importance of reading and education. I remember being extremely bored in kindergarten - 4 or 5 years old - and getting told to sit in the back with a few other kids who could read already and we'd read books while most of the other kids were being taught to read.)