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.
No-one besides the OED.. There is no central authority for English, but if there were one it would be the OED. Besides, there are copious examples of people using it in that way in English for half a century: the OED catalogues these so it's not really up for debate in English. And that's the point, it's English not Yiddish we are talking about. You would say "it's Yiddish, not German" if a German started trying to correct you on things because he's a "native speaker".
Edit: as far as I can tell the article always said "originally a Yiddish word" which serves to hold the case I am making.
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.