The thing is that "potato" is used here as a mass noun (uncountable) that describes the kind of stuffing. I believe that "potato" is the mass noun and not "potatoes." When you say "I'm eating dumplings with potatoes," it sounds like you have dumplings and extra potatoes on your plate. However, I'm not a native English speaker, so it would be nice to have a third opinion on this.
"Potato dumplings" sounds to me like the dough is made of potato. If I google "potato dumplings" I get recipes of dumplings that are made of potato dough, such as German Knödel or Polish kopytka. Ukrainian varenyky are made of wheat flour, like Polish pierogi, or like Italian ravioli (but larger and shaped differently), or like German Maultaschen (but shaped differently).
20th century Canadian Ukrainian did not call them "varenyky”, but something like "peraghe"or "peroghy". Most of those people came from Galicia, and that dialect strongly influenced the way Ukrainian was spoken in Canada for several decades. I never heard dumplings called varenyky by my Canadian relatives. And they were always з картоплею whenever I had them.