Queen, gavra, obviously I’m making a point. The reason Chazal said to teach to swim is because it’s dangerous to go in the water. It’s also dangerous to go to the library.
Yes, the child can walk into the library on their own. They can also walk into McDonald’s on their own. If you allow them to go to the library, you are essentially being mechanech them that the library is a kosher place to be, and if not, that it’s not a kosher place to be.
Will they listen? Hopefully, if you have a good relationship with them, and they know your Yiddishkeit is sincere and real, they will want to please you and emulate your values.