Icimoi, yes the child may have other phone numbers in her phone. All well and good if she has been told that these are the people to contact if she has need and they know she has been told that and are available to help if the need arises - not okay if these are just numbers of people who have no idea they might be contacted.
The mother leaving her number with the child is absolutely NOT the same thing as leaving it with the school in case of illness etc. if the child is ill at school, the parent is contacted but has the reassurance of knowing that until they can leave work or can get to the school, a recognised adult is in charge and looking after the child. If the child is in the library alone, this is not the case - the child is alone,without a specific known adult they can turn to, until the parent can get there.
Workplaces accept that employees might sometimes have to leave to collect a sick child etc from school. They should be able to reasonably assume assume that parents have put some kind of childcare in place and that their employee won't need to leave to deal with matters other than illness - with the child in the library left alone for half a day, there could be all kinds of reasons why the child felt the need to speak to the parent, or why the parent might then feel the need to go and see them - not on the same level as illness, but requiring a bit of adult attention anyway.
The fact that the parent says she is flexible in work and can nip out if needed to, just isn't good enough - it means the child has to be left in the library, so that 'nipping out' is viable because of the distance, and it quite simply is not good enough, because actually she is working and unless it is a job where she is totally flexible about when and where she works (clearly not,with a rigid 3 hour working slot) - she needs to accept that if her daughter is too young to be let at home for 3 hours, she cannot be both the child carer AND at work at the same time.
The OP wants too much. She wants to be at work and for her child to not be at home alone. However the OP clearly can't be in 2 places at a time or do 2 things at once and during this tricky age until the girl CAN be left alone, she really needs to put something in place which involves someone specific who is available.
And finally, I really don't think what the OP is proposing is akin to a child travelling to school alone. Firstly, this is perfectly normal - journeys are not usually 3 hours in one go, are often with other children, and those running public transport at school times do keep an eye out for children and have the contact details of the schools in case of need. Children making such journeys can contact a parent or they can contact the school - the school will be available to offer advice or to make contact with a parent if necessary - the child is not left alone. Crucially though, we are not talking about 3 hours at a time for 10 year olds, even if over a period of a week, the total amount of time spent travelling is more than 3 hours or even 10.