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The author is not stating this is a 1st Amendment violation. She is stating that the act is an encroachment on her own freedom of speech - by the paper, not the government.
This is what the author says: "This is a serious, blatant and outrageous encroachment on freedom of speech and no newspaper in this country has the authority to limit editorial or opinion content based on their own PERSONAL set of opinions." For you to say the author isn't claiming a 1st Amendment violation by the newspaper is ridiculous, since there exists no other "freedom of speech" outside the 1st Amendment. A newspaper gets its authority from its owner, and no one else.

Again, I will cite that this particular publication invites submissions from their readers for republication. Picking out only those that reflect their opinions is almost a form of censorship and it's damn dishonest reporting at very least. The Times invites readers to voice their opinions, yes? So, why do they censor those they take acception to?
The letter writer is specifically asking the newspaper to put something into its "reportage," and offers herself up to be interviewed. That is completely different from its opinion section, which invites comments from readers. If she writes a letter to the editor that meets the newspapers posted guidelines but still doesn't get printed, she might have a case. If the paper doesn't print any letters from people with the same opinion she has, she would have a huge case. She so far hasn't provided any evidence to that effect.