There was a short period when it was
actually open for the 2000 Holy Year, at least for small guided tours. Now it's closed again on a more or less permanent basis since 2005. I have always lived in Rome
and I remember since I was a young boy that the DA was always "closed for restoration".
I don't know if it would actually be feasible to uncover the underground parts, they were used as foundations to later buildings and the large environmental change could easily damage paintings and structures.