Burnside isn't close to being the most incompetent general of the war. You want utter lack of military ability, try Ben Butler. Or Gideon Pillow.
He had some notable successes, which got him promoted beyond his ability. That's how he got the chance to make such massive mistakes.
And in fact, there were occasions when Burnside could have gained major successes. The Crater, for instance. If Meade had not interfered, the attack probably would have succeeded, leading to the destruction of the AoNV, the fall of Richmond, and the end of the war in a few more months.
OTL, Burnside remained popular after the war, as shown by his role in the National Rifle Association, and his service as Governor of Rhode Island and US Senator. With a major victory to his credit, he could go farther.
One difficulty is that he died in 1881 of heart failure. If (as OTL) Grant is elected in 1868 and re-elected in 1872, Burnside has only 1876 and 1880 as possible opportunities. Also, his being from a small state is a problem...