In the introduction (section 1.3) of their paper D-modules on the affine Grassmannian and representations of affine Kac-Moody algebras (http://front.math.ucdavis.edu/0303.5173), Frenkel and Gaitsgory give a new proof of part (2) above. Their idea is to factor $ \Gamma : D^\lambda_{G/B}-mod \rightarrow U\mathfrak g - mod $ in two steps: $$D^\lambda_{G/B}-mod \rightarrow U\mathfrak g - bimod \rightarrow U\mathfrak g - mod$$ where the first functor is given by $ \Gamma(G, \pi^* \mathcal F) $ (using the projection $ G \rightarrow G/B $) and the second functor is given by $ Hom(M(\lambda), M )$ where $M(\lambda) $ is the Verma module.
In this way, $\Gamma $ is the composition of two exact functors and thus is exact.
I don't know how this compares to the original proof of exactness.