When I speak ill of the concept of third party presidential campaigns solving anything, someone inevitably brings up the precedent of the Republican Party. Quite clearly, Abraham Lincoln and his administration solved quite a bit. This is, in my view, the exception that proves the rule in the true sense of the term.
The crux of the matter is that the GOP, when founded, was a movement of party-switchers from among the ranks of established politicians. Galusha Aaron Grow, the first Republican Speaker of the House, is a case in point. As a young Pennsylvania lawyer, he ran for congress as a Democrat in 1850 and won. He ran for re-election in 1852 and won. He ran for re-election again in 1854 and won again. Then he switched parties in response to Franklin Pierce’s signing of the Kansas-Nebraska Act, became a Republican, ran again in 1856 and won again. Then he ran again in 1858 and won yet again. Then he ran and won in 1860 and became Speaker. Charles Sumner was appointed to the Senate by the Massachusetts State Legislature in 1850 and he sat as a Democrat, espousing anti-slavery views until nearly beaten to death by (fellow Democrat!) Preston Brooks. While convalescing back home, he was re-appointed by the legislature in 1856 this time as a Republican. Abraham Lincoln served in congress as a Whig, was an Illinois Whig activist and 1854 Whig Party Senate candidate, then ran again two years later as a Republican.
Which is all just to say that what happened in 1860 was not at all the case of an outsider third party presidential campaign sweeping the nation and changing things up. Instead, starting in 1854 and with continuing force in 1856 and 1858 a large number of established northern politicians left existing parties and came together at the Republican Party. Then, with caucuses already in place in the House and the Senate and strong bases of support in every northern state legislature, they won a presidential campaign against a splintered Democratic Party. So, yes, a third party that manages to persuade large numbers of incumbent officeholders from both parties to jump ship and join it could have a huge practical impact on American politics. But this simply underscores the fact that a lone wolf presidential bid, even if somehow successful, would change little. You walk before you run. You win seats in congress and identify a regional base of support before you sweep the nation. Presidential elections are fun. Party building is tedious.