My first thought was that this is all totally reasonable, but for someone so close, it would have been better delivered personally, as part of a conversation.
However, I agree with those who say that the update gives us more context, and honestly (because we can be on here) I think your sister unreasonable.
If that is how she reacts when having a conversation, I am not surprised that a very clear email was sent.
I would have hated, as a new mum to have been required to call by someone's house - and neither my mum or MiL would have expected it. Your niece sensibly realises that she may be in discomfort, if not pain, with an unpredictable new baby; she doesn't want to add in a 'complication'.
We don't know, but now have a glimpse of the relationship. I suspect your niece is anxious not to be overwhelmed. I don't know, but I wonder if your sister hadn't pushed so hard, but graciously accepted the initial request, then she might not find the lines being drawn so firmly.
Because I'm a 'pleaser' I might have said 'OK' to calling in, then rung to say 'didn't call round, not feeling up to it'. However, I applaud your niece for being clear!
PS: What would your mother have expected of you and your sister - where you supposed to dance to her tune? I would have thought any woman who has given birth would realise that she want to get back to home comfort with her baby asap.