Most hunter/gatherer or agrarian societies in the world, as well as most of the urban ones until quite recently, have kids running around naked or lightly clothed, just pooping when they need to go. It’s actually dramatically more hygienic for the kids than leaving poop in their diapers for long periods of time (or in whatever other pants/underpants, in times and places where diapers were unavailable), not to mention it wastes a lot less resources.
I agree though that having kids poop in the street stops working as well once you have very high population density, and I can imagine how it would be shocking to people living in repressed societies who are afraid of their bodies, and I can imagine how it would stoke xenophobic anxieties about new immigrants.
As for what your parents can remember about their ancestors: I suspect that their impression of historical realities is severely skewed. My parents are anthropologists, so I grew up spending lots of time hanging out with rural peasants in mud houses with dirt floors, right next to the edge of the forest. But in talking to my own aunts, uncles, grandparents, etc., basically none of them have absolutely any conception what life was like for their own ancestors in the late 19th / early 20th century, who were rural European peasants who immigrated to the US to live first in squalid immigrant neighborhoods of eastern/midwestern cities and later kept going westward and settled on farms. If I try to explain to my grandmother that her own parents basically grew up in a house pretty similar to peasant houses everywhere (rough hand-made construction, no running water or electricity, animals all around, etc. etc.) she finds it basically inconceivable. As in it’s literally impossible for her to imagine how much different her own life was from that of her parents and grandparents.
Which is to say: unless your great-great grandparents were nobility (and probably even then), I’m pretty sure they ran around ****ting all over the place as kids.