As an anti-free-speech progressive, I'll give you an answer: it's
not the dark side of that proposition. It's the dark side of assuming that it's possible to trust corporations to be fair about free speech at all.
The only reason there was a hue and cry about corporate censorship of, say, Donald Trump was that Donald Trump's ideas were extremely popular and a lot of people had already heard it, and the people who wanted to hear it have significant (though not absolute) political power.
There are people with nowhere near as much political power who get censored, and the same folks cheer it on. A few years before the recent debate, the corporations all stopped hosting the free speech of sex workers, and nobody from any party cared, and nobody even thought of mentioning it during the more recent debate. The problem is that "free speech" got redefined. The words of the president of the US were within the pale; the words of sex workers were not. Which of those was actually at risk in the marketplace of ideas?
It's abundantly clear that saying that corporations are required to host speech they want to will simply require them to host certain politically-correct types of speech that the government recognizes as "free speech".
We're deeply familiar with censorship on the left. We experienced the House Un-American Activities Committee. To this day, you cannot become a naturalized American citizen if you have been publishing or circulating "written or printed matter advocating communism." Why should we believe that supporters of "free speech" are advocating for anything other than the legal right to force people to publish their own speech?
Show me a free speech advocate who is arguing for the repeal of FOSTA/SESTA, the repeal of the anti-Communist provisions in US naturalization law, the repeal of gag rules preventing pediatricians giving their honest medical opinion or obligations for gynecologists to say things they don't believe, or relevant to this thread, the various anti-BDS legislation that's been passed recently, and then we'll talk.