Bad counterexample? CloudFlare did just that, triggering a strangely pro-censorship round of hacktivism in the form of #OpISIS.
https://www.theguardian.com/technology/2015/nov/19/cloudflar...
Moreover, how is DMCA relevant? Copyrighted works are outside the bounds of free speech.
DMCA, and in general the US legal system are extremely relevant to me as a user. If I have a theoretical right to free speech, but in practice any big US media company could kill it then I'm much better off in another country where maybe the theoretical right is 10% less but I can actually practically enjoy that right.
In the USA, Neonazi publications are allowed, while in Germany they are illegal. Therefore, the US has a freer press than Germany. Freedom House would disagree, and they are simply wrong.
The US ranks lower than Germany in the FH and RWB freedom of the press indices [1], which while not quite the same is highly related to freedom of speech.
[1] https://en.wikipedia.org/wiki/Censorship_by_country