Mini BX is full of super cool ideas, for sure.
My “overflow damage” mechanic is a sort of half measure of this, since I wanted characters in this to be a bit more durable than the average OSR/NSR character. However, I agree that it’s not super intuitive, and ends up being a bit awkward in practice. I’m not sure I want to go all the way to letting damage move all the way down the chain, but it’d probably be worth testing at least. Or at least including as an optional “higher lethality” rule.