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I don't think it had anything to do with the Nazis, as far as the origins of a Kindergarten. or the word itself. It went back farther than that.
You have to figure a situation, going way back, where people lived close together in a village but worked in fields outside of the village. Houses in the village were built with common walls, with no yards, etc. but certain areas inside of the village were left natural, like a park (Garten). Some areas were also left natural, near the city wall, for growing gardens, too. Also a called a Garten.
Kids too young to go out into the fields were taken care of, as a a group, by a designated adult, and they could, in good weather, gather in ones of those park areas, to play. So, Kindergarten. They weren't for indocrination, rather, just a way to take care of young kids while the parents were in the fields.
I did have a neighbor who was a young girl/tween under the Nazis. Yes, they were indoctrinated, but I would say by extensions or re-purposing of structures that were already in place. There, the Nazis could use the existing Kindergarten system to indoctrinate, for example. Hitler Jugend, an exception, there, I would guess, though. I think that was a Nazi structure from the ground up. Still know some former Hitler Jugend.