I agree, mostly. And this is such a kind post. I had no family help at all, and I think as a result I'm struggling to imagine how the OP could be so sick that she needs her mum's help, but also not so sick that her husband isn't taking her to an urgent doc appt. But then, I'd want better for my own kids - which your post reminded me of.
But, then, I also wouldn't accept, or offer, a formal 8-5 childcare arrangement from, or for, anyone, and at the same time be beholden to a relational ideal where we help each other out as and when. I genuinely don't think you can have it both ways.
I do think @Mkxxx you are being unreasonable. From your posts, I get a sense of entitlement to your mum's time, which probably arises from your mum's generosity and the lovely relationship you seem to have with her. I get that it's hard, but if you messaged the night before, then you didn't know whether you'd be able to mind your kids or not. It was pre-emptive, and it's fine, in that case, for your mum to have said, sure, once I've been to the gym. Do you appreciate that? I'm not saying you're a brat, but I'm not wholly agreeing with you either.