I think most Elsies were originally Else, or had a mother or grandmother named it. In Germanic languages the final E is not silent as in English, so you have to spell it Elsie for English people to pronounce it approximately right when they read it off the page.
Elsie would be a completely wrong way of transcribing Else in most Germanic language.
Elsa would be closer.
Or “Elseh”.
Elseh I'll grant, but it feels a lot less natively English, and you can't count on an American clerk or census taker to care about accurate transcription. It may have been closer to "that's not a name in English. Pick either Elsie or Elsa".
And yeah, Elsa was common too.
Elsie sounds more like a German diminutive or pet name of Else than a strict transcription.
Elsie was quite a popular name in the 1920s in the UK and US - the 65th most popular in the US [1] in that decade - so I think it's very unlikely most of those 42k Elsies had some relationship with an Else.