No, the "relaying" or "transmission" argument is just a dishonest gambit in 2018. Only idiots believe YouTube thinks everything some YouTuber might say on their own channel. Should AT&T have had the right to cut off phone calls containing speech they didn't like? There were party lines where entire groups of people could hear the same person speak, so it wasn't just one on one communication. Should AT&T have had the right to cut those off because of AT&T's "free speech" rights?
Sorry, but the effect of such a position is to allow large companies to regulate the speech of the general public. Their inability to censor the general public doesn't somehow mean that they lose any ability to express their views to the public. It only curtails their ability to squelch the expression of others.