Preface - I'm not sure that import diagnostics and Feature Works exist separately, hence my following instructions.
Do you have FeatureWorks? If so, don't run Import Diagnostics when you import the file. Exit out of it, then run FeatureWorks. It will delete the surface bodies that are causing you the issues, and then it'll recognize the features giving you a model tree. If you don't have feature works, You can simply shift select all the surfaces, RMB->delete bodies. You'll then end up with a single dumb solid.
Running FeatureWorks (exiting import diagnostics and selecting YES to feature recognition) results in an error that states; "Part contains features with rebuild error. FeatureWorks could not proceed further."
However your second suggestiong did work for me. After deleting all of the surfaces, I ran import diagnostics. This again produced the errant faces, but they were easily healed. As you state, I now have one big, dumb, solid - suitable for my immediate needs.