NYT is responsible for anything they write. NYT is not responsible for anything that some random person writes in the comments.
No law or regulation compels any private entity to host any speech that they don't want to. There are no "open forums" in private spaces. An act of Congress can't undo the protections of the First Amendment.
A single entity can be both a publisher and a platform; nearly every news/info site is. Any article that allows third-party comments is functioning in both roles, from NYT to Fox News to Ammoland. The article is publisher content, and the comments are platform content. Platforms can be moderated and still have liability protections re: third-party content (comments, user posts, etc.).
NYT would be responsible for something that random people post in the comments if they remove points of view they don't like while leaving points of view they do like, for example. At that point they're effectively "publishing" the content.