The thing is, with a child of 10, however grown up and happy to be in the library, how certain can anyone be about how they will feel for half a day left alone?
Many people will leave a child of that age alone at home or somewhere like the library for half an hour. Sometimes even the same child enjoys being left and other times doesn't.
Personally I think half a day at age 10 is too long, whether at home or in a public place. The OP just cannot know what might happen (especially in a,public place) to throw the child out of her comfort zone. However mature and confident the girl is, there is no knowing this. An older child does just need to cope with some of that stuff, but I honestly think 10 is too young to put a child in the position where that could happen over 3 hours or more, which is why the OP doesn't want to leave her at home - she isn't ready for that length of time.
In my mind, this means she isn't ready to be in a public place for so long either, when the only adult available to her is at work. Yes adults expect to leave work for emergencies SU habs sudden illness of a child and employers expect that occasionally. Is it reasonable for an employer to be told that their worker might have to pop out, because that day, she is actually the childcare - so it might not just be for sudden illness, that other childcare such as schools manage until the point when a parent can arrive, but the OP is the only childcare that day. If I was the employer, I would ask or wonder if this child is beyond having childcare now, so that is why the OP has not provided any.....but that is not the case.....this girl usually has childcare, but the OP is choosing not to have it on these days.
It think it is wrong to think it is okay as an employee to need to provide more than emergency care whilst at work (and the kind of help the girl might require whilst in the library is probably the care a proper care-giver could offer if present at that point - not the type that would require a parental summons in an emergency) and I think it is wrong for a 10 year old to be left in that position without access to an individual who really can only be available when there is an emergency.
Yes I want my children to grow in independence. This is being carefully structured and built up, with adequate support in the background. It is led by their needs, not my requirements to work or lack of third are. In this case, I really don't think the support is adequate or that someone working should be providing it - it is a conflict of interests for someone with a child of this age.
For an older child, I think this scenario would be fine. Older ones need to negotiate difficulties and manage longer periods alone. But for a 10 year old to be left for BOTH 3 hours AND with very limited proper support, just seems to ask too much of them. Never mind the library staff etc....I think it places too many demands on the child.