Yes, when my DS was 9 he was left at the library for programs and activities. Now at 11 he is allowed to bike to the library by himself. (almost 2 miles each way- this is the distance he would need to travel to get to school if not homeschooled). DS started taking classes at the library around age 5-6 and I would stay in a different part of the library until he was about 7 or so. From 7-9 I would vary between drop off and staying. Now I am fully comfortable doing drop offs but I like to stay because that hour gives me time to browse the shelves, read, do paper work etc. However I have no issues with DS in the library etc.
Our library has a sign up that says 'children under the age of 9 must be accompanied by an adult'. This means you can not drop your kiddo off for the day etc. The librarian is not a babysitter, however you do not need to stay for programs etc.