Absolutely! And in my opinion, the reason this didn't happen also has to do with one major flaw of the implementation of what we got to know as "socialism", namely the centralized, top-down planning of all aspects of life. Of course that won't leave room for much creativity! But the comrades were so afraid that imperialistic elements would infiltrate into their art, that they felt they had to keep a very close eye on what they would permit.
Meanwhile, a look at the hotbed of capitalism (US and UK) in the early 20th century should have told them the whole story: real imperialistic music can't hold a candle to the passion filled music of the exploited, such as the Blues, including all its off-shots. Even then, once the 1%-ers discover it for themselves, as it was with early Jazz, by the time it leaves the smoke-filled bars of the hood it becomes watered-down elevator music. The same thing can be said about Rock'n Roll or Hip-Hop.
So what I believe should have been an amazingly fertile ground for communist tunes, that truly appealed to the masses, were 19th century factory towns, not even in Russia so mcuh, but more probably in Germany or France. I bet that's where real fiery music was created! Lots of it may have gotten lost, though I also think lots of it I just haven't found yet on my musical exploration.