A duel is an artifact of a bygone age that views dishonor as worse than death. Matters of honor, and following through on promises, sometimes don't directly make sense in a purely pragmatic sense. People dueled to settle matters of honor as recently as the 18th century. Alexander Hamilton, the guy on the 10 dollar bill, died in a duel with political rival Aaron Burr. It made sense to the countless individuals who've fought in duels through the ages. Read some history.
Soarin made a promise. He gave his word to Baron Redtail (under a bit of duress, admittedly) that he would face him in aerial combat. If your promises are only as good as personal expediency allows, then your word isn't worth much. Soarin's engaged to the Element of Honesty. As AJ said in the scene in the Salt Lick, no fiance of hers is gonna go back on his word, no matter what. She was prepared to mourn and avenge him if he lost, but if he'd run off or gone back on his word, he probably wouldn't still be engaged to her.
Long story short, promises aren't always easy or wise to keep, but if you have a personal code of honor then keep them you will.