Claisen ester condensation: what's the point of the second deprotination, and then reprotination?

The deprotonation at the end isn't intentional. It just happens by itself, because 1) you're running the reaction in basic conditions, and 2) the hydrogen alpha to the two carbonyls is fairly acidic. And, yes, it does help drive the reaction forward, since the resulting negatively charged product is stabilized by three resonance forms. We use the acid workup afterward to neutralize the product and any remaining base floating around.