You may want to revisit your history books. Or do you think that the progressives at our universities that steal or burn campus newspapers and intimidate both speakers and audience members do not abridge first amendment rights? Or are those thugs merely exuberantly expressing their rights to free speech?
But A&E having a contractual relationship with the Robertsons may have perfectly legal constraints regarding what happened. I don't know any of the terms of contract. That said A&E I suspect is not only cowardly but terminally stupid.
Read the 1st Amendment again. It only proscribes the government from suppressing our right to free speech. It does nothing to compel an ordinary citizen from suppressing that same right.
It's become an article of faith in America that the 1st Amendment compels private individuals and groups to respect the God-given right to free speech, but it really doesn't.
Come into my house and start saying things I don't like, and you'll probably be asked to leave. I don't think you'd argue that I have a perfect right to do that. Free Republic does it all the time, as is their right. Is that suppressing free speech? No, it's not. It's simply reacting to speech they find repugnant, and that's the price people pay for speaking their minds in a private setting.