Earlier quoted context omitted.
but it inevitably breaks down at some point because I haven't accounted for something along the way Could you give more details? That might be helpful in providing some ideas.
For example, I might have instructed them to roll for three character attributes. And then they encounter an enemy which has hit points of some sort and maybe attack and defend numbers. And then I realise that something about their character attributes doesn't really gel with how I'm trying to get them to roll dice against that enemy. Or, wait, they need to jump across a pit but did one of their attributes apply (str…
Forgot a certain ability on your streamlined ability sheet? Well, who says they all have to be determined beforehand like in DND? Just make them roll new ability specs just when you need them.
Or simply make due with what you've got, e.g. by adjusting what you originally had in mind for the enemy attack with what is possible within your simplified system. And for the next game, slightly update the system whenever you've identified an oversight.
Don't sweat it, the exact rules are not so important, as long as your players can rely on consistency (as a proxy for fairness). For kids, I think, in the end they're gonna love it when they can just roll some dice every now and then and observe how what their characters do results in success when they roll well and (non-catastrophic) failure when they don't. Other than that, I would imagine the theater-of-mind aspect, i.e., the plot and the general world setting will probably be much more important.