> French cooks served roasted pigeons and Ukrainians and Russians found
more affordable variant of the dish and named it “holubtsi.” Do you
think her story is true?

That’s what Russian wikipedia article states without providing references. The name little doves or little pigeons are also used for the same dish in Slovakian, Polish, Lithuanian cuisines.

There are numerous variations of the dish in Ukraine depending on the region and season. In some regions of Ukraine beetroot leaves are also used.