We see that in Exodus 19:3-6. There the Lord makes his covenant with Israel and says that by entering into the covenant (rather than because of any presumed purity) they will become a holy nation (goy qadosh). The Hebrew phrase translated "holy nation" in Exodus 19:6 could also be translated "a holy people," and by extension "a nation of saints," in other words a people covenanted to God.
Just as holy objects are those set aside for holy purposes, the saints are the people set aside for God's holy purposes. Presumably those purposes range beyond the purposes of the individuals who make up the nation, and presumably those purposes are not God's recognition of the moral superiority of those in his nation. Indeed, the Hebrew Bible goes out of its way to regularly show us people who are fallen and yet nevertheless the chosen people.
Mormons have no reason to assume that what was true of ancient Israel, true of the New Testament Church, and true of the people in the Book of Mormon is less true of ourselves: We are covenanted to God to be tools for his work, and by virtue of that covenant we (among others rather than instead of them) are chosen people. We are called saints. The covenant that makes us saints also obligates us to be pure. But it is no guarantee that we are.