In the 1837 edition, the subordinate conjunction that in this sentence was removed, probably because the intervening parenthetical “were it not for this” made the occurrence of the following that awkward. Without the parenthetical clause, there would have been no motivation for deleting the that. There are two other places in the text where a parenthetical statement intervenes between “I say unto you” and the following that-clause. In neither of these cases has the conjunctive that been deleted, perhaps because the that is immediately followed by a subordinate as-clause:
The critical text will restore the that in Mosiah 15:19, despite its difficulty. (Another grammatical emendation would be to place the that before the parenthetical statement: “I say unto you that— were it not for this—all mankind must have perished”.)
Summary: Restore the subordinate conjunction that in Mosiah 15:19, despite its difficulty: “I say unto you—were it not for this—that all mankind must have perished”.