The thing is, the SIL doesn't even sound particularly anxious.
I'm the least anxious person. I've been lucky enough to really enjoy the baby period. And if anyone (but especially a rather resentful, jealous, judgemental SIL who was clearly just itching for the chance to bustle in and show me how UK care for my own baby) wanted me to leave my baby with her when small, I'd have refused. I didn't leave mine with anyone when tiny - that's just the way I wanted it.
And the calling. I'd certainly call, and do, as do all the parents of the children (family and nursery mates) that I've babysat for. It's normal to check in, as much a 'how you doing, hope all's ok' for the benefit of the babysitter as much as the child. IT'S NORMAL.
Your posts OP make it clear that you're the kind of parent who is quite happy to have a tiny baby babysat by others as 'it's good for them'- lots and lots of mothers would really disagree (esp if breast feeding). Neither way is right - the only right thing is to remember that everyone does it differently and everyone parents their own way. The biggest sign of an overly 'anxious', insecure parent is one who starts squealing and judging and huffing when someone does it differently to them - they feel their choices are bring criticised.
This is you. You don't like your SIL. A part of you feels anxious and annoyed that your brother chose someone so different to you, and now they are parents, a different style of parenting has arrived in the family and YOU feel judged. That's really the issue. It's why you so studiously don't mention your brother in this - you want this to be all about SIL, you want to see her as the problem, the interloper, the one not doing it as We Do In Our Family. Ooooh, she's soooooo anxious! Silly cow!
Get this: the choices they make are as much your brother as they are her. Your brother parents differently in HIS family. You will have pissed him off as much as her. Carry on the way you are, and you'll soon see where the family boundaries lie: their family will distance themselves from yours, and this child you so patronisingly 'see almost as one of yours' will grow up hardly knowing you.
And as for trying to 'teach her a lesson that she can trust you', err, by not returning a call and being out of touch when caring for THEIR child - Jesus, that would be IT for any question of trusting you in the future! No, you aren't trustworthy. You're pushy, judgy and feel you know better about their child's care than them. I wouldn't trust you to care for my child, because the main thing is that I feel HAPPY that my child is with someone who respects my parenting. So while I'm sure they're upset about the situation, I'm also sure that there's beena conversation along the lines of - that's it, she's not having her again.
Also. Has it occurred to you that your brother and his wife very likely might not be huge fans of the way YOU parent? To you, it's a case of 'her' (of course) being 'wrong' as they're different to you. To them, it's just as likely that they think you a bit careless or hands off. But, as they're NORMAL, they wouldn't say that. You simply don't comment on other peoples parenting styles - it's very bad form!