My assumption was that the destruction of so many ships created a situation where battlefield promotions of young officers who had proven themselves would be possible.

It's not without precedent in US military history:
http://en.wikipedia.org/wiki/Galusha_Pennypacker
http://en.wikipedia.org/wiki/George_Armstrong_Custer
http://en.wikipedia.org/wiki/Battlefield_commission

Still, I agree that it would have been more likely that Kirk would have been promoted but not given command of a starship immediately. But hey, a little dramatic license is okay with me.