Certainly there are forms of restricting someone's ability to be heard that would infringe on their right to free speech, but the former does not necessarily imply or require the latter.
However if I don't want to hear what you say--or I don't want to help you say it--the government is not going to step in and force me to do it.
The distinction is really about depriving vs. providing.
People get it mixed up in their heads when it comes to contractual relationships like Cloudflare. On one hand it is grammatically correct to say that Cloudflare is "depriving" 8chan of their service, since 8chan used to have access to it. Legally, though, 8chan never had any right to Cloudflare's service. It was a privilege that was revoked.
Correct. However, the main/only service CloudFlare provides is preventing DDOS attacks, which fall plainly into the category of unconstitutionally depriving someone of their free speech.
Providing services to 8chan isn't furthering their mission in any way, and the only "advantage" of cutting them off is that it empowers the people who would deprive the 8channers of their constitutional right to free speech.
The comment I was replying to was about what the constitutional right to free speech entails, not about what CloudFlare is required to do.