Isn't "everybody can succeed with enough determination and work" the American Dream in a nutshell? (not talking about how it works out in practice, since that's not the point in your "soviet era" scenario either)
The American Dream of success is different. In the US, for various reasons (probably including mobility & lack of local extended family) part of the dream is to have Mom stay home with Kids while Dad works 40 hrs/week to afford the House, Car, and TV. Those are different contours than the more communal ideals espoused in the Soviet Union or even most of Europe today.
