Well, there's definitely a few reasons, but they can be summed up pretty succinctly with:

1. Davis was already acting President by the time Breckinridge left for the Confederacy.

2. Breckinridge didn't leave for the Confederacy immediately, only running in September of 1861; 2 months before the Confederacy's presidential election.

3. He was made a brigadier general a few days before the election.

So, why did these things matter?

Well, in the case of 1), we can say this contributed heavily to the choices. While Breckinridge was urging compromise and crying out against Lincoln's war policies (he was still a Senator for Kentucky, after losing the presidential election), states were already seceding. Jefferson Davis was the pick for acting President until the election in 1861, and changing presidents that early on seemed to be a pretty futile effort. It would be similar in a way to the FDR-related "don't change a horse mid-stream" argument.

When he was to be arrested for disloyalty in the Union (he was also trying to keep Kentucky neutral), he fled South before he could be caught. However, he never really believed the South could win, as he confided in friends in Frankfort, KY before he fled. He still entertained some hopes that a united bloc of slave states could get Lincoln to compromise, and keep the conflict from continuing.

That sort of covers 2), as well, though as you can see 2 months isn't much time to curry favor when you've already got an acting President who's already been in power for 8 months, is experienced in war (Davis was Secretary of War for 4 years), and you're nothing but a lawyer/politician.

Still, Breckinridge was made a brigadier-general (as I brought up in point 3)). Why? Well, the idea was initially just to get his enormous prestige in Kentucky to be a boon to the Confederate cause, by rallying Kentuckians to the Confederacy. Breckinridge was, though, a surprisingly adept military commander. He was described as "aggressive, dynamic, [and a] leader". Still, he had a huge crisis of conscience in that regard: it's been said that he couldn't take cases in which he had to prosecute as a lawyer because of his fragile conscience. He would only act as a defense lawyer. (The idea here is that he couldn't handle having to confront someone, though he could defend others when necessary). Anyways, he would often be seen crying over fallen comrades after battles, and walking with tears down his cheeks after battles. It's even suggested that the stress from the experiences and the way the faces of the dead haunted him for the rest of his life contributed to his life being shorter, though that's hard to justify since he died of what was recorded as cirrhosis (liver) and complications.

Once Breckinridge proved himself on the battlefield, there was no reason for him to do too much in the political sphere, though he was appointed Secretary of War near the end of the Civil War for the Confederacy. Still, while he may have been the best choice for the South in peacetime, to try and keep control of the executive branch, he was not nearly experienced enough in matters of war, nor around enough at the start of the conflict, for anyone to consider him a Presidential candidate.

As for his prowess in war, yes it was unexpected, and some examples of his exploits include:

• His first battle was at Shiloh (Union victory), and while that went poorly, he learned many lessons that he used to his advantage in the future. He would be promoted to major-general, the highest position given to so-called "political generals", because his personal performance was great (he led reserves).

• He fought at Vicksburg, and led his first independent command in an attack on Baton Rouge, and he actually took Baton Rouge, though it couldn't be held.

• He commanded a division in the Battle of Stones River at the end of 1862. Despite the battle going poorly, Breckinridge performed well...and was right about the attack that he was ordered to undertake was stupid.

• He directed repeated assaults that paved the way at Chickamauga for the most complete defeat of Union forces the Confederates would get. He broke the Union line, though he wouldn't destroy their army entirely.

• He led an army (that he raised himself) to victory against the odds at the Battle of New Market.

• He then went to go meet up with Lee at Cold Harbor, where Grant's heaviest losses would take place, and where Grant would be halted.

While not the best ever, for an untrained lawyer with no military experience, he certainly proved himself. He would command in numerous battles that the Confederates were doomed to lose, but his men loved him and he did well with his portions. Still, no one expected that, so no one was ready to put him in charge of something so important as the military endeavors of the Confederacy.

Sources:

John C. Breckinridge William C. Davis The Register of the Kentucky Historical Society, Vol. 85, No. 3 (Summer 1987), pp. 197-212

JOHN CABELL BRECKINRIDGE: Lawyer, Statesman and Soldier of the Confederacy GEORGE R. FARNUM American Bar Association Journal, Vol. 29, No. 5 (May, 1943), pp. 270-271