You might be able to center up off the countersink around the holes where the stud is broken below the surface.
Be aware that a properly centered hole in the middle of a broken off screw thread does not 'look centered' because the helix of the thread is coming up one side and tends to make you think the hole is 'over here' when it really is not.
Honestly, I would not like to take the approach of drilling the stud out completely. I would drill a small hole right through, using a small drill press because I want to feel when the drill enters the clear space between the stud and the bottom of the original hole.
Spray some good penetrating fluid down the hole. Maybe let is sit overnight. Then try the easy out method. It may still not work.
Ultimately, I would drill maybe a 3/16 or 1/4" hole and use a small welding rod and stick weld down the hole, maybe 1/4" deep or so. I use a hi strength rod, UTP65, for this, available in 1/16" dia. Just weld around in the hole, and get it good and hot. The weld will shrink and pull the stud with it and break the rust bond. Continue to peck weld and build up the stud until it is as high as the top of the hole. Then, weld a little piece of keystock to it, put the vise grips on it and wiggle it out.
This method will leave the original threads intact.
First you get good, then you get fast. Then grouchiness sets in.
(Note: The opinions expressed in this post are my own and are not necessarily those of CNCzone and its management)