Drills love to wander and make oversized holes. Drilling manually makes it worse. I do not like wonkie stuff on my builds. IF I was drilling a tang manually I do it like this:
Make the punch mark with caliper scratch marks and a prick punch. Then check the mark carefully with digital calipers. Adjust it if necessary by steering the cone to the direction needed. Then drill undersized. Check the hole with digital calipers. Correct the side to side with a file or carbide bur. Once centered, clean it up with a larger bit. IF it is now perfectly centered, drill to size. I never drill an important hole in one shot, it is to risky.
Or, use the milling machine.............
On the OP's question, I'd file it to make it centered. Then make my piloted screw chamfer to fit the oversized hole. IF necessary I'd make a screw to fit the hole. I doubt that is needed. The head is all that you see. IF the hole for the stank is a bit large it does not matter.