For the most part, we're SUPPOSED to believe that Starfleet people are a cut above average to begin with. Officers, anyway - most of them go through a greuling application and testing process before they even get to the academy. Point being, that between hundreds of episodes of Trek it's easy to assume that most officers are pretty brilliant to begin with.
So, Bashir pulling incredible cures to alien diseases isn't THAT much of a stretch, nor should it be. What *I* mostly remember of his early years was more his cocky nature and social awkwardness, which is perfectly natural for a young and brilliantly nerdy Doctor who spent much of his last decade in classrooms or med school. You don't need to be genetically engineered for that, and I'm sure that's what the writers were thinking when they retconned his DNA into the show.