[DebugInfo] Remove unnecessary flushes and add additional testing

The flushes previously existed to help ensure consistent error message
output when stdout and stderr were passed to the same location. This is
no longer necessary as errs() is now tied to outs().

Reviewed by: dblaikie, MaskRay, JDevlieghere, labath

Differential Revision: https://reviews.llvm.org/D80803