The popularity of the term “self-reliance” goes back to 1841, when Ralph Waldo Emerson published his book of the same title. Emerson was a strident individualist, and to embody his philosophy, one must be true to their own heart and mind, independent of popular opinion and social pressure. A commonly quoted phrase from the book is that a “foolish consistency is the hobgoblin of little minds.” I imagine that Emmerson thought Mormonism was just such a thing.
When early church leaders did speak of “self-reliance” it was not unalloyed. For example, in 1927 Apostle Orson F. Whitney spoke in General Conference, stating: “Self-reliance is a good thing, if not carried too far. But self-assurance, self-sufficiency, self-conceit, is a bad thing. There is no such thing as absolute independence. We depend, upon one another, and all are dependent upon God.”