In the introduction (section 1.3) of the paper
- Ed Frenkel and Dennis Gaitsgory, $D$-modules on the affine Grassmannian and representations of affine Kac-Moody algebras, Duke Math. J. 125(2) (2004) pp 279–327, doi:10.1215/S0012-7094-04-12524-2, arXiv:math/0303173,
they give a new proof of part (2) above. Their idea is to factor $ \Gamma : D^\lambda_{G/B}\text{-}mod \rightarrow U\mathfrak g\text{-}mod $ in two steps: $$ D^\lambda_{G/B}\text{-}mod \rightarrow U\mathfrak g\text{-}bimod \rightarrow U\mathfrak g\text{-}mod $$ 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.
