I am no expert but I have drilled a zillion holes in thing's, that's my career.
IMO drilling on the lathe is the only way to go. The benifits are you can let the plug rotate on the drill bit every few seconds by letting go of the grip slightly and see if it is drilling straight. The most important thing is the start of the drill. If the thing is not dead nuts, machinist lingo, from the start than it will probably not turn out good. Use a short lenght drill so it is ridgid to start and go on from there. Also make sure you clear the drill bit of chips every few seconds. That is another reason for run out. The heat builds up and the chips clog the flutes with nowhere to go, thus the bit wanders. If I ruin 1 in 30 I get pissed. Practice makes perfect. Good luck.