I also work in a public library, at the reference desk. My name tag says "librarian", though, technically, I am not one yet, still in grad school. I also deal with the children that some parents leave here, on a daily basis.
It is not an issue of being able to deal with people. I like working with people - helping the patrons find the information they seek - I enjoy my job. But I am bothered by parents who leave their children at the library for hours at a time.
It's a simple issue of safety and responsibility.
Most of the parents who leave their children here wouldn't think of leaving them alone, unattended in other public spaces like, for instance, the park, but they feel fine leaving their children here.
The library is a public space. All sorts of people use the library. Most of them seem to be good people. But there are also some that definitely seem a bit creepy. I am sure that this is the case everywhere. Parents wouldn't leave their children alone with these people normally, but they seem to feel comfortable with it at the library.
The library, unlike public schools, is not in loco parentis. If I saw something bad happening, I would do what I could to stop it, of course. But much of the time, I am occupied by other work. Additionally, there are large parts of the library that I cannot see from my desk.
Leaving young children alone at the library for long periods of time is the same as leaving them alone elsewhere - it's abandonment. Though I doubt it would happen here, I know that at some of the libraries my classmates work at, they have a policy of calling the police to report an abandoned child if a young child has been left there alone for more than a certain period of time.
I try to explain this to parents, generally to their surprise or even annoyance. They seem to equate the library, as a place of learning, with being a safe place for their children. I'd argue it's a good place for their children, to encourage learning and study, but that that does not make it a day care center.
I am not trying to discourage children from using the library. I am not one of those grouchy librarians who hates children. I'm always happy to see them using the library. But it does worry me that parents think that we will watch over and care for their children.
Maybe there is a need for publicly funded child care. I don't know. I'd like to think that there's a more cost effective way to do it than hiring a bunch of people with master's degrees to work there.