I would definitely do this on the lathe if the holes need to be centered. You can do that on the mill with a v block, but I think it would be easier on the lathe. An ER or 5C chuck would easily work for what you want, so it will come down to which one you would prefer to have. Instead of a length stop in the chuck, you could use a caliper or some sort of repeatable stop to set the amount of workpiece sticking out from the chuck so that you can quickly repeat the operation on each piece and not have to change settings every time. You will get bored doing that many pieces, so the easier to do it and avoid mistakes is the way to go.
I would definitely get spiral flute taps for what you are doing, they are so much better than hand taps for blind holes that you won't want to use the hand taps for that ever again. I find my spiral flute taps cut much better than my hand taps, so it makes it less tiring to do multiple holes.
I would definitely get spiral flute taps for what you are doing, they are so much better than hand taps for blind holes that you won't want to use the hand taps for that ever again. I find my spiral flute taps cut much better than my hand taps, so it makes it less tiring to do multiple holes.