Specifically on freedom of speech, the left and right oscillate tons based on who is feeling more authoritarian (the other axis). A small, very vocal portion of the identity-politics left is currently very authoritarian and anti freedom of speech but I think that's a specific current internet-influenced trend and not to do with a wider shift of the overton window on issues in general.
I agree with you on the internet-influenced part, but not on the latter portion of your argument. Even if the anti-free speech movement is small, the majority of "left-aligned media" do echo their beliefs. This to me does swing the overton window to the left.
