If clearances are done properly and unused pads deleted, there should be no exposed copper except for the part of the barrel that wasn't drilled out, even without filling the hole. Here is some information from our manufacturing group.
Backdrilled holes can be backdrilled before filling (no exposed copper), or after filling (exposed copper looking down on a cross-section of the barrel of the hole), or on non-filled holes. That means that they could be drilled before filling which means before the second drill for the unfilled holes (drill the identified back drill holes, plate, backdrill, fill, drill non-filled holes, plate again) or after second plate (drill the filled holes, plate, fill, drill the identified back drill holes and the non-filled holes, plate, backdrill the back drilled holes), or they could be back drilled at the end of all the processes if no holes have filling (drill all, plate, drill the back drilled holes).
There will be over size clearances for plane layers and there will be pad suppression for signal layers, so the only exposed copper is the "top" of the barrel. That should not be a problem in most applications and so back drilling (stub drilling) is not itself sufficient reason to incur the cost of via filling. But if vias are going to be filled, it makes sense to fill the holes but only before back drilling. Filling after back drilling has it's own special issues.
Identifying the back drilled holes in a separate drill file or set of files, and indicating back drill on a drill chart, automatically, from an input say in the design of the pad stack, seems to be missing and would be nice to have. Right now we have to add all these notes about what layer not to breach and how deep is the hole (or conversely, what is the maximum remaining stub) by some special designation in the hole chart that points back to a sheet 1 note - having logic in the design tools to do this would be a plus.