So how about this... What if, rather than drilling a preliminary pilot/centering hole, I mount the center punch in the drill press, and with it turned off and using my alignment jig, put the the center punch dents into my wood that way?
Aye, thatd work, probably. You wont have any rotating forces trying to push the punch somewhere else, but theres still the risk that the quill doesnt always come to the exact same spot.
If youre already using a jig, might not be a bad idea to rethink how the jig works. By the sound of it, your jig is secured to the table and just used to register the work piece in the same spot, yes? What i would do would be to grab a piece of sheet good, something like MDF, and drill the holes in that as accurately as possible. The, take that piece and secure it to your workpiece with the holes where you want to go and use the holes in the template to guide the drill. This way, even if the quill wanders the drill itself cant go anywhere, since the jig is keeping it in place.
Picture something like a dowel jig for ideas:
A bushing guides the drill, so even if the tool is inaccurate the drill can only go where you want it