From a marketing perspective, however, I don't think adopting the 'socialist' moniker helps the (American social democratic) Left, or any other group in American politics really.
By using the label of 'socialism', they allow their critics to invoke Mao, Stalin, Chavez and Venezuela, etc. Basically the totalitarianism of the state-run command economy.
I think they would be better off selling their program as a more 'humane' form of capitalism, or--better yet--reclaim the term 'Free Market'. After all, when Adam Smith and the classical economists talked about 'free markets' they meant markets free from the rentier class--markets in which prices would be brought in line with the actual cost of production, rather than artifically inflated by the parasitic, unproductive costs of interest on predatory lending, monopoly rent and land rent.
Since the right has largely become the defenders of the rentier class, and perverted the term 'Free Market' to mean a market free for rent extraction and stacked against labor and industry, the (American social democratic) left has room to bring the term back to its original sense and use it as a marketing tool.