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.
absolutely disagree in this case. if NOBODY is going to libraries and my kids do, that may be a different case. But when i walk into a library with my kids and i tell them to sit on the comfy chairs while i look around for books for them to choose from, and that they are NOT to go browsing thru the shelves, but they are welcome to go to the librarian to ask for specific books that they have already gotten approved, there is no message of permissibility there. not only that, but they learn very clearly that the library is NOT a place for browsing independently and after years of that training i dont need to worry about them one day falling into the place and getting sucked in.
Some places need to be off limits, and some places cannot be off limits so the children need to be taught how to stay away from things they will encounter in these places. Same goes for malls, department stores with HUGE televisions playing, grocery stores with magazine racks etc.