That's a pretty big and misleading stretch. The Democrats and the Republicans 'swapped' in the 20th Century, so of course they don't have history to show for it, but that swap was related to the 'New Deal' after the Depression, which brought about a more 'left-wing' social contract that we would associate more closely with European/Commonwealth Democracies. From Wiki (my bold):
I don't mind being wrong, as I'm not a strong scholar on this, but here's
more evidence for strong unions in this time from The Atlantic:
So the seeds were there for the way America would become. Still, Roosevelt's vision was successful enough that it created a consensus, and it can still be said (as Wiki comments), that the power break-down in that time was very much Democratic:
1933-1945 Franklin Roosevelt Democratic - congress with him for basically all of his 12.1 yrs. He then died.
1945-1953 Harry Truman Democratic - congress with him for 6 of 7.8 yrs. There were complaints about strikes (labour power) early in his time in charge, though, and he became anti
some Unions. Congress went Republican in 1946 (Nixon and McCarthy were freshmen), and passed the Union-limiting 'Labor Management Relations Act', which Truman vetoed, but Congress overrode that. Democrats won it back in 1948 with Truman saying he would repeal the act, but southern Democrats didn't allow it. That's a familiar problem still stopping progress on issues in today's America - 'the South', politically, are fairly right-wing. Truman's 2nd win was a surprise. Democrats even tried to recruit a replacement for him beforehand. That man was:
1953-1961 Dwight Eisenhower Republican - congress with him for only 2 of 8 yrs.
1961-1963 John Kennedy Democratic - congress with him for all of his 2.8 years. He was then assassinated.
1963-1969 Lyndon Johnson Democratic - congress with him for all of his 5.2 years.
Nixon is then elected, showing how the narrative of America politics had gone in a different direction. Tricky Dicky was a paranoid guy, and keen to tell Americans they too were being left behind by others (Goldwater saw the power of that narrative).
But the Democrats still had both houses, and would do so up until Reagan's election, when we saw that 'neo-liberal' thing really take hold in America. The Democrats would then not win the Presidency for 14 years, until Clinton did his 'centrist' pitch, which as we know, was still corporation-friendly. As with all the 'Bernie would've won' claims, people willfully ignore the political reality of America (more often that happens in order to pretend the Democrats are similar to the Republicans). The Democrats didn't win power until they moved closer to the Republicans. They did more for the poor (like Obama did), but they didn't legislate a drastic change to America like Roosevelt did in the New Deal and like Reagan reversed in the 80s.
The narrative may have changed again now, thanks to Bush's incompetence and the GFC highlighting just how much America's economic policies were failing (at a macro level to match their failures at a micro level for so many), but the vote for Trump was still significantly right-wing in nature, so we can't be sure yet. It looks like Don Don has done enough to ensure the narrative has shifted, but of course his fans and the right-wing media will continue claiming otherwise (e.g. their gun control propaganda), so it's hard to predict. Information warfare is a real issue here.
This is already too long, but anyway, here's a quote from Eisenhower which references 'labor laws', to show where he was coming from as the Republican in that 36 year run listed above. It also shows that back then it was Republicans having to accept the political reality of American voter's opinions:
Also, for what it's worth:
tl;dr Democrats do have history of being pro-working class people.