If there really was something in Ellie's brain, couldn't the doctors have experimented with a dead Infected, or try to trap one? Wouldn't her brain have infected the doctor, if he'd started the surgery? Was it only through biting that you could be infected? How much of a risk was Ellie to be around?
shareEllie couldn't infect other people, so no risk at all.
As far as experimenting on infected goes, if the doctors before the fall and at the start couldn't figure out a vaccine with all their resources, I doubt one rando in a filthy post apocalyptic hospital will figure it out.
I'd wager that even with Ellie, a viable vaccine was just wishful thinking. Good call Joel!