Problem is holding the pinion concentric. With a worn three-jaw, plus locating on teeth, the hole is unlikely to end up in the middle. A home-made split collet can fix that. And a worn lathe probably doesn't have an accurately set tailstock, hence suggestion of a boring tool. It's easy to start over-complicating, isn't it? It's what I would do, but I have the tools to hand. There are probably folks out there who would hand-hold the pinion and use a drill press (but please, don't try that one at home...)