I haven't done a massive study of the subject but most other Christian denominations that I'm familiar with do not place emphasis on Jesus' prayer that he offered in Gethsemane (relatively speaking of course), Jesus' sacrifice takes place exclusively while on the cross. For some the atonement began in Gethsemane, for others on the cross. Depending on which atonement theory you subscribe to there's a good argument to be made that Christ's entire life was an atonement.
I'm not sure what other denominations make of the event in Gethsemane, likely the agony Christ suffered was the result of the realization of what was to shortly take place. The LDS version of the atonement greatly expands the event in Gethsemane. In some ways we view that event as the defining moment of the atonement. There aren't many scriptures that spell out what we believe happened in Gethsemane. We have:
(D&C 19 uses similar language)Mosiah 3:7 wrote:And lo, he shall suffer temptations, and pain of body, hunger, thirst, and fatigue, even more than man can suffer, except it be unto death; for behold, blood cometh from every pore, so great shall be his anguish for the wickedness and the abominations of his people.
Which ties the Gethsemane event to suffering because of the wickedness and abominations of his people. It also uses language that makes bleeding from every pore more literal that the language found in Luke. There really aren't many points of origin for the theology that Jesus atoned for the sins of mankind while in Gethsemane. For me the theology flows better when the price for sin is paid on the cross.
There's the concept of a blood atonement. Maybe it's the sweating, as it were, drops of blood that is the source of the Gethsemane atonement theory.
This isn't a problem that needs solving, my views on the atonement are very different anyway. I was just curious about the history of the LDS version of the atonement, how we got to where we are today with relatively little scripture to support our unique stance. Was it a latter-day prophet expounding on a verse people found to be mysterious, did the teachings solidify during a time when leaders were distancing themselves from using the cross, etc.?