“The South chose a path of violent secession and lost.”
Except for the “lost” part, it sounds like the Founding Fathers.
Now in the truest sense, I have no dog in this fight (my ancestors immigrated to the US after the Civil War). However, I am from the North (PA). The whole story is much more complex than you would make it out to be—there were mixed motives all around, with some of the "villains" having their moments of decency and honor, and some of the "heroes" having their moments of treachery and inhumanity.
Lincoln certainly had the right to see both Davis and Lee hanged... but reconciling a broken nation required the magnanimity of welcoming them back as brothers and fellow Americans—from Lee and Davis all the way down to every ordinary farmer who donned the Confederate gray. Lincoln let them keep a shred of their dignity even in defeat. Lincoln's magnanimity allows us to see him as one of the great leaders of history (and allows us to overlook some of his more unfortunate views/decisions, such as the suspension of habeas corpus in Maryland and his racism toward blacks).
Rubbing a loss into the losers' faces rarely endears them to the winners, and more often than not, leads the former losers to rise up again (and if you need an example, look at the way the Germans were shamed by the French actions in the Rhone Valley following WWI, and how that contributed to the rise of Hitler). Hanging Davis and Lee would have made some folks feel better for a time, but it would have run at cross-purposes to Lincoln's primary goal: preserving the Union. That is far more than mere "political expediency."
Finally, If I were visiting the Jefferson Davis museum, even I would wear the hat. The haters need to get over themselves.
I would say you need to do a little more research. None of what happened is crystal clear.
Lincoln opened fire on April 6th when he sent a fleet of warships to attack the confederates surrounding Fort Sumter. What did you expect them to do? Wait until the cannon balls of his ships were killing them before they did anything?
Lincoln sent the war fleet to kill. He fired the first shot.