When men are depressed and struggling with their kids the general advice on here is that, along with counselling/support, they need to do MORE with the children, not less, so that the bond can strengthen.
Does your sister work op? How are workloads like housework, child drop offs/pick ups, bed times split, if you know?
This won't be a popular opinion but I know lots of mum who say awful things about their kids, are obsessed with naps to get a break etc. I genuinely think too much is now expected of mum's and we're starting to see the affects... So many work demanding jobs then come home and still do all house stuff, most of the day to day parenting, appointments, planning etc and it's just too much. Downtime is mostly spent scrolling rectangles of anxiety that companies have spent billions making us all addicted to. People are burning out and just don't have the capacity.
You see it all the time on here, thread after thread of how people hate parenting, countdown to bed time, are more shouty than they want to be, dread weekends with their children. And it all gets fed with people agreeing how shitty everything is, how little kids are just knobs and to just hang on until they're older. Which I get because people make those threads when they're already feeling down and people relate and want to offer solidarity, but I'm just not sure the relentless negativity helps in the long run.
I hope your sister agrees to get the help and support she needs op. You sound like a lovely, caring person.