Second, we've lost some simplicity: Whereas the auction format only amounts to setting a reserve price, the desired equilibrium now depends in a detailed way on the type distributions.

On the latter, it's useful to think about the "must-sell" case, where a 2PA with no reserve is optimal in the symmetric regular case whatever the type distribution. This argument says it's still optimal in the symmetric 2-player case, but now the equilibrium depends on details.