One of my biggest problems with human nature, in general, and the way that members, in particular, tend to interpret things is the tendency to read something applied to one thing or person and extrapolate it to everything or everyone else.
Let me give two examples to illustrate my point:
1) Oliver Cowdery was told how HE received a witness from the Spirit. HE felt a burning in HIS bosom, and HE experienced a stupor of thought in HIS mind. That passage says absolutely nothing whatsoever about how anyone else will or should recognize the influence of the Spirit in their own life - nothing, at all. However, members and missionaries alike tend to promise others that they will feel it in that exact same way. I rarely feel it that way, and I know others who rarely feel it that way, but thousands of people are being told that is how they can experience it. That really is a HUGE concern and pet peeve of mine.
2) Moroni 10:3-4 promises those who remember, read, ponder and pray that God will manifest the truth of the Book of Mormon unto them - but it says absolutely nothing about how that will be done - how any particular reader will experience that manifestation. It also does NOT say in those verses that everyone who reads will "know" it is true; all it says is that the truth will be manifest somehow. One person might feel it is true; another one might sense it is true; another might have a classic Cowdery burning; one might have a powerful dream; one simply might feel compelled to continue reading it and be drawn into it inescapably; one might experience it in a way that is unimaginable to you and me. Only one of these "manifestations" match Oliver Cowdery's method of manifestation, but each and every one is "real" to the person involved.
President Hinckley didn't say, "Every single thing taught in the Church either is 100% true or else the Church is a fraud," nor did he say, "The Church is never wrong or else it is not of God." What he said, in essence, included three foundational claims:
1) Joseph spoke with God (somehow);
appeared to Joseph, who "translated" the Book of Mormon (somehow); and Moroni
3) the Priesthood authority to perform eternal ordinances was restored (somehow).
Either those three things happened (again, somehow) or they didn't. Either heavenly personages appeared to Joseph (somehow), or he made it up. They either are "real" or "fraudulent". Joseph either believed in his visions, or he lied about them intentionally. He either was sincere, or he was a fraud. That's essentially all Pres. Hinckley said in the quote.
To me, the crux of his comment is that either Joseph was telling the truth about the biggest things (the heavenly manifestations), or he was a fraud. Either he believed it, or he faked it. I can accept that, since I couldn't remain faithful if I was convinced he lied about those foundation things. There are lots and lots and lots of things that I believe are totally open to ambiguity and mistakes; the heavenly visitations/visions aren't among them - even though I think there are multiple, legitimate ways that those visitations could have occurred without being fraudulent.