Probably the major thing that holds the American Left from wider adoption of their alternative name however is the legacy of the last major American political movement to use that name -- the "Progressives" of around the beginning of the 20th century. Let me note what an embarrassment THEY are:
Like all Leftists, they were a fractious lot so there is no fixed list of what a Progressive believed but here are some of the notable features in Progressive doctrine: They were (of course) anti-business, pro-welfare and believers in central government management of the economy but as well as that they believed in eugenics, racial superiority, war as a purifying force, nationalism, imperialism, conformity, discipline, book-burning, loyalty oaths and straight-armed salutes. And they despised the Jews, of course.
Does all that somehow ring a bell? Where else have you heard of such a set of ideas? You've got it! All of those ideas were also Hitler's ideas. And THAT'S WHERE HITLER GOT A LOT OF IT FROM. The American Progressives were the first Fascists of the 20th century. In America they were mostly unable to put those ideas into full practice but Hitler showed what happens when you do.
Which in part explains why the present day "Progressives" are so desperate to hang on to control of the public schools. I mean gee, gosh, and wow. What would happen to the Left if the schools actually taught the real history of the last century?
A round in the back of the head in the old Soviet style?