She has a MIL, so it's possible she is not a single parent, although I still refer to ex-MIL as MIL. Regardless, it's pretty clear that at the moment, she wanted her child to be with her MIL during this appointment. It's possible that she didn't expect to be home until perhaps 5pm or so and didn't want to impose on friends locally or it may have conflicted with local friends' schedules.
I told her that I had an appointment and I wasn’t available to collect my daughter; she then said what appointment is it, why have you got it, when did you find out about it.....so I felt that I shouldn’t have to disclose that information as it made me feel uncomfortable.
I don't feel the receptionist would need all this info - she has no need to know what the OP's appointment is, why she has it, or when she found out about it. "A medical appointment" is really all she needed.
I think that in the push for 100% attendance, schools have gone too far - we've had a few instances where school said our child should have attended school. For example, (as stated previously) when they were ill (school said we should let THEM decide if they were well enough to attend school, which is utterly inappropriate). Another instance was the school telling me that we needed to schedule all medical/dental appointments outside of school hours. I have two children with disabilities - their specialists send appointments by post and that is almost engraved in stone. You can attempt to change the appointment - but then you may be waiting another month or two to get in, which is not a good option. And some specialists only have "clinic" appointments on certain days and hours, so you take what you get. As a parent, I want my child to be educated, but it is MY responsibility to decide if they are able to attend school - not the school's.