And failing to free her at all may cost him the opportunity to save the world. Harry should have had some doubts as to whether he was ready for the mission.

Failing that, the other thing that has been bothering me for a while is why did Quirrel take Harry to save Bellatrix now? If Quirrel was pure Voldy he wouldn't care about Bellatrix, he doesn't love her. Saving her now, by taking a young idealistic boy on an important high-stress mission, doesn't seem like a good plan. How much does an evil overlord value saving henchwomen, what risk is worth it?

I am not sure that Quirrel is pure Voldy. I'm half tempted to predict that Quirrel is Harry-grown-old-and-dark transported through time in some fashion. Hence the extreme inability to touch each other and the fact that Quirrel's priors are too good. There is a fair amount of evidence against that (lack of patronus, for one). But it is a fun idea.