As a six year old in First Grade my reading ability was ahead of at least most of my classmates. In fact my First Grade teacher sometimes had me help other students who were just learning to read.
And yes, when we were taken to the school library by the second semester, every child looked through the available books and chose what he/she wanted to read. We lived in a small town and by the time I was eight years old I was also walking by myself to the city library and checking out books. I was a voracious reader back then. Still am.
We were limited to two books each check out and of course had to return the books we already had. There was a fine involved that came out of my teensy allowance so I was very responsible to get the books back on time. It made for a lot of trips to the library.
But most of America shared solid values of propriety, what was or was not appropriate for children to say or do or have. I am pretty sure any sexually explicit material was kept well out of reach of minors in both the school and public library and probably had to be requested by adults.