In a one-page tract, "A Brief Outline, Of the Faith & Doctrine of the Latter Day Saints" from 1844, we have the following affirmation that Christ atoned for our sins by means of his crucifixion, showing that early Latter-day Saints (here, the year of the prophet's death) were not averse to appealing to the crucifixion and its centrality to Latter-day Saint soteriology:
The first principle of theology as held by this Church, Father in God the Eternal Father, and in his Son Jesus Christ, who verily was crucified for the sins of the world, and rose from the dead on the third day, and is now seated at the right hand of God as a mediator, and in the Holy Ghost, who bears record them, the same as to day as yesterday, and forever. ("A Brief Outline, Of the Faith & Doctrine of the Latter Day Saints", as found in A True and Descriptive Account of the Assassination of Joseph and Hiram Smith, the Mormon Prophet and Patriarch At Carthage, Illinois, June 27th, 1844, By an Eye Witness, T[HOMAS]. A. LYNE [New York: C.A. Calhoun, 1844], 19, emphasis added)
For more, see:
John Hilton III, Considering the Cross: How Calvary Connects Us with Christ (Salt Lake City: Deseret Book, 2021)