No need to drill out any position markers, and for the love of god ... DO NOT USE CHEMICAL STRIPPER!
If your guitar has any sort of binding, chemical stripper will melt it right off!
If it doesn't have binding, it'll still melt your position markers.
Use sandpaper to strip the finish ...
Then tape off the fretboard, including the edge (or binding) which hold the position markers.
Once you paint the neck, remove the tape on the edges of the fretboard (or binding), leaving the tape only on the face of the fretboard. Then proceed to clearcoat the neck and the edges of the fretboard.
Once you're done, simply wetsand and blend everything, then compound and buff.
I'm thinking your goal is to refinish, not rebuild, right?