Ask yourself this: if he'd just said there's no need for all the shouting and not offered to do it on Monday, what do you think the reaction here would be?
Dollars-to-doughnuts, it'd be "He's a lazy shit, just mansplaining on something he can't be arsed to do, divorce him".
He's putting his money where his mouth is, rather than just lecturing you on how you should do stuff. That's not a good reason to be an arse with him.
Whether he's right or wrong, there's actually probably a decent middle position you can arrive at. It's clear that it's not doing you any good to have the ridiculous stress in the mornings, so maybe there's a way to better train the kids to be cooperative? Perhaps some "persuasion" from him might be the way to get there; since they're used to giving you crap in the mornings, and your reaction to it, your shouting likely means nothing to them any more purely through familiarity.
Kids are, in their natural resting state, unpredictable and manipulative little shits. They may well behave themselves perfectly on Monday, though, purely because it's a sudden change and they don't yet know how far they can push him.
Full disclosure, I don't know - I never had a herd of children in the mornings, just the one (and she was nightmare enough when she wanted to be). On the occasions that I saw my wife losing it in the mornings, though, I took over the next one and...child was good as gold, butter wouldn't melt etc. It was quite maddening, if I'm honest.