Firstly, well done! It sounds like you've managed a massive turnaround and that is no mean feat.
For my part, though I don't claim to be an expert I would generally assume that whoever holds access to a bank account in a child's name is technically holding that money on legal trust for that child. I'm sure that someone will correct me if that is not necessarily the case, but I think it's a useful moral guide anyway.
My rule of thumb as a parent would be to act as "trustee +" for my child, as it were. So my absolute preference would be to never touch that money receipt to invest it for them as best I could do that they have it for their own use when they become an adult.
The other thing I would carefully consider using it for is an opportunity that arose that would significantly benefit my child long-term (such as an amazing education) but isn't something that we could afford for them to do ourselves. I'd check into the legalities of it but that's something I would seriously consider if the aforementioned was then losing out on that opportunity. I'd pay it back over time too.
In terms of emergency family money, I personally wouldn't include it in my calculations, even "at the back of my mind". I'd treat it like it didn't exist for the purpose of deciding how much emergency savings I built up. So in that sense I'm with your sister.
In practice, if we were genuinely in a position where we would, for example, all lose our home if we didn't use the child's money as a stop gap (and there was no other option) I would check what was allowed and then do it on the basis that it keeps a roof over their head and that I would prioritise paying that back too. I wouldn't use it to buy us an extra month in our home if all was lost anyway, if you see what I mean.
I would also rather take out a loan than use it to mend the roof or whatever other day-to-day emergency because it's just not my money. As soon as you start expanding the categories of scenarios in which you might use it that aren'tdirectly for your child's benefit, I think you wrongly blur the lines between what's yours and what's theirs.
So, yes, there are very limited situations in which I'd use it for my child's benefit, but I don't think it's an answer to your sister's criticism to point to your child's money.
As far as I'm concerned, you have 4 months with of emergency savings (and, again, well done you!)