I tried this method tonight. I turned about an inch or so round. Mounted the blank in my collet chuck. Then I chucked up my bit and slowly drilled away.
EXTREMELY successful!!
Items of note...
1) My bit never got hot. No heat makes a clean hole and no drift.
2) The natural method of advancing the tailstock meant that I HAD to drill slow. This also contributed to previous point
3) the divots from turning it round acted as a great guide so the bit didn't wander.
Yes, drilling on the lathe is more tedious and takes longer. However, the result a markedly better and worth the effort. I don't think if you are slamming together a slimline, it is worth it, but there are lots of cases when it is. I just figured out how to make mechanical pencils from deconstructing Pentels (pops wanted a 9MM mechanical pencil), and WOW, I wish I had thought of drilling them on the lathe.
Thanks for the advice!