Reb Moshe Feinstein was once interviewed by the New York Times where he answered questions like how does Klal Yisroel know who is a Gadol? Does anyone know when it was, and does anyone know where I can get an archived version online if someone can link it It would be a greatly appreciated or even how i could go about getting it?
Many library systems have access to PDFs of the historical New York Times. The 1975 interview is an article from May 5, 1975, entitled: "Responsa: The Law as Seen By Rabbis for 1,000 Years", by Israel L. Shenker; pages 33 and 61.
I can see the PDF but as it's owned by ProQuest, I don't think I'm allowed to repost it. There's a picture of Rav Moshe answering a phone call in his study.
The article starts with some material about Rav Moshe, noting the old refrain that a new rabbi would get an ordination certificate and Rabbi Feinstein's number; it also mentions a volume published opposing him (ma'aneh le'igeret) and "handbills of defiance" in Satmar for one of his rulings. The article then goes into describing what responsa literature is and some of its history, then some description of Rabbi Ephraim Oshry's Holocaust responsa; then has some quotations with Conservative and Reform responsa experts on their views.
The only direct quotes from Rav Moshe in the article are, as you'd indicated:
You can't wake up in the morning and decide you're an expert on answers. If people see that one answer is good, and another answer is good, gradually you will be accepted.
A rabbi who replies to people's questions works harder than a doctor dealing with a case of life and death. The doctor is responsible only to the patient, but the rabbi is responsible to God.
His obituary is March 25, 1986. He is described as:
... a towering scholar whose rulings guided the religious practices of millions of Jews around the world. ... Rabbi Feinstein, a short, richly bearded and well-groomed man ... His answer was usually accepted as law. ... His mystique was based on an intelligence and a command of Halakha ... that other rabbinic scholars found awe inspriring. He was said to have studied the 62 volumes of the Talmud ... more than 200 times ... Yet rabbis were also struck by what they described as a remarkable humility.
The original article is available online: http://docslide.us/documents/ny-times-1975-article-about-r-moshe-feinstein.html
Here's a quote from the 1986 New York Times obituary about R' Moshe:
Typical of his modesty was this answer he gave in a 1975 interview as to how he had acquired his standing as a posek.
"You don't wake up in the morning and decide you're an expert on answers," he said. "If people see that one answer is good and another answer is good, gradually you will be accepted."