As a native Yiddish speaker, my experience is that schlep is only used to describe a journey-related pain in Yiddish - I think TFA statement that this is a Yiddish word used in this context is what's confusing (has the article been edited to clarify this since?). It appears that its use has been expanded by non-Yiddish speakers to refer to "arduous tasks" - although no-one besides OED seems to think this.

As it happens, in Yiddish, there is a one other use which is more related which is to describe someone as "a schlepper" (or that they "schlepped") which means they take a long time do something.