Obviously, in the 20th century where there's discontent or potential for disruption you'd find socialists or Marxists or communists or fascists, so clearly if you look at grievance movements (racial minorities, feminists, gay activists) you'll find leftists involved in the early stages.
The phrase "political correctness" was used by Communists in the 30s and by the New Left in the 60s before it was popularized by leftists, feminists, gay activists in the 1980s. You certainly could say that the word has a history involving Marxists.
But European Marxists didn't develop what we now call "political correctness." That was something Americans came up with and the connection to Adorno or Horkheimer is pretty tenuous. It's like blaming the philosophes for the French Revolution or Nietzsche for Nazism. There are intellectual connections, but they aren't as direct as some people suppose.
Lind seems to think that if he applies the phrase "cultural Marxism" to political correctness often enough and loudly enough, political correctness will just go away, and that won't happen. There are real political interests involved that don't derive from European theorists.
Finally, if the phrase "political correctness" has a dubious ideological history, so does the phrase "cultural Marxism" (Kulturbolshewismus). It's not really something one should throw around as freely and as naively as Lind does.