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](https://doi.org/10.1215/S0012-7094-04-12524-2), arXiv:[math/0303173](https://arxiv.org/abs/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.