If you’re saying that “Duty” in the sense of “poop” comes from Latin via Old French, that’s something I’d like to see a cite or evidence for. I’m not aware of any use of it earlier than the introduction of “duty” in its other sense. And, contrary to what you’re saying, it seems to me altogether likely that this is a euphemism, especially given the “England expects Every Man Will Do His Duty” – pooping being the baby’s duty.
I’m saying I don’t think “Doody” (faeces) comes from “Duty” (as in “Obligation”) and I’d be interested to see a cite to the effect it does. It could equally be argued the word comes from"do" or “doo-doo” because that’s what babies do - poo a lot.
Euphemisms were a Victorian thing - earlier eras were typically more direct except when it came to political satire or diplomacy. And in all the Victorian writings I’ve read, I’ve never seen a reference to babies “doing their duty” by virtue of nappy-filling.
It sounds like one of those myths about Victorian prudishness (like the little skirts on furniture legs) that people keep repeating because it sounds good and has a high truthiness quotient, but isn’t actually verifiable (or has even been shown to be incorrect).