I do all my drilling after turning. I may lose 2 plugs per 100, maybe, using the lathe. I am not a fan of brad bits. Waste of $$ IMO. Start off with a short drill, both ends, and let out more drill length but no more than is needed. I go around 1200 rpm's. I use a 5/32 drill. I use a drill with a very short flute length,aircraft drills, like an inch of flute. This gives you rigidity and that is the key . A long fluted whipy, flexible, drill will walk all over the place. You should be relieving the drill, chips, about every 1/2 to 3/4 of an inch of depth, hard woods even less depth. When I have an off center plug I drill from the rear to the belly hole closest the lip and then drill from front to the belly hole by hand to match up the thru hole. Remember this is just my opinion.
Either way you should rarely get a screw up using the lathe for thru drilling. Good luck and more practice. P.