8, 9, 10, 11 - it's still young and they do need a bit more attention than your average user and because I'm human I would feel a bit more responsible for a child that age in my branch than I would a capable adult.
But there are loads of people who need more attention than the average user. That's what "average" means. When I was in the library the other day, there was a man who seemed very unfamiliar with computers trying to use one, and the librarian was spending quite a bit of time with him. He needed a lot more attention than your average user. Should he have been chucked out?
People keep saying that OP is planning to use the library staff as cheap childminders: she has made clear that she isn't. She doesn't expect them to watch her child any more than they would any other library user, and she expects her child to sort herself out with getting books, using the computers, and going out to the café etc. Staff have a duty to supervise all library users. If they feel the need to have a quick look at OP's daughter occasionally, how is that any different?
People also suggest that three hours is a long time for a child to spend reading. I am one of those who could and did easily spend that amount of time buried in books as a child. But, again, even if this child doesn't want to do that, she has the facility to use the computers, go out to the café, play games on her phone, or indeed bring in paper and pens in order to write or draw pictures.
It's worth drawing a comparison with the time children spend in travelling to school. By law, the maximum travel time should be 1 hour 15 minutes each way. At secondary level, most children go to school on their own. Therefore it is deemed entirely acceptable for 11 year olds to be out on their own in public on the roads, buses and trains for 2.5 hours a day. 3 hours in a library is an easy option by comparison.