The great key to understanding the 1860 election was that the Democratic Party could no longer be united because of slavery. The various events of the last several years had reached a point where northern Democrats were no longer willing to defer to southern Democrats on the issue of slavery, and compromise was no longer possible.
You had the northern Democrats who supported Stephen Douglas who supported popular sovereignty, but who would not force slavery on those territories which did not want it.
You had southern Democrats who wanted absolute protection of slavery's future and expansion. They supported John Breckenridge.
Finally, you had those people who hoped the entire conflict could be resolved by keeping things exactly as they were, and they supported John Bell. This group is often associated with southern Whigs who weren't Democrats, but who couldn't support the free soil Republicans.
All major political parties are actually coalitions of different interest groups who merely find agreement on most things and are willing to compromise in order to win elections. Eventually, those coalitions find the political situation has changed and that they no longer have enough in common to continue to work together. Historically that marks one of the great political realignments in American history, which have had several of them. The Democrats simply could not agree anymore on the slavery issue, and were bound to disintegrate.
The Republicans though were entirely united on the slavery issue. They were completely against its further spread, although its members disagreed on the issue of abolition. Lincoln was chosen as the nominee because he was, in fact, a moderate who could have broad appeal.
The 1860 election showed that a candidate no longer needed to win the south in order to become elected president. That's what prompted the fire eaters to secede. Any Republican who won would have caused that reaction.
But no matter who the Republican nominee was, the Democrats were no longer capable of unity. Southern slave owners wanted to make radical changes in policy to keep their cooperation. Unless northern Democrats, who were the majority of the party, were going to completely let a minority seize control, the southerners were going to walk out.
Furthermore, if you look at the electoral map, each of the candidates basically ran separate campaigns that appealed to a very small sectional group. Douglas's votes were in the north, and Breckenridge and Bell split the vote in the South (which Lincoln was never going to win anyway). Even if everyone who voted against Lincoln voted for one candidate, that only meant Lincoln would not win CA and OR which together had 7 electoral votes. So Lincoln would still win the electoral college by 173 to 130. Of course, he'd lose the popular vote.
But this would never happen, because the Democrats cannot unite. The northern and southern Democrats could no longer agree on slavery. Any candidate acceptable to one group is not acceptable to the other.