Just some food for thought for you, I have been turning this over in my mind. I'm not thinking about carburetors, more the re-jetting of a natural gas burner to propane. Same process, different size drills. In the first place, putting a "pin vise" in the tail stock chuck is not arbitraily on center, it will need checking and truing before use. In the second, small drills need to run very fast at slow feed. Using the tail stock feed, look for broken bits. And often botched parts. I have occasionally mounted the body in a lathe tail stock chuck or drill press chuck. But centering the bit takes a while. Usually I work that small with brass. Steel would be a different matter.
Most times I will drill such holes by hand. The handheld version of a "pin vise" is regularly used by hobbyists and jewelers. The ones I have are by General and X-acto, a little under 10 bux last one I bought. These two versions have a double ended body, thumb swivel on one end, collet on the other. Two double ended collets handling Nr 80(.0135") to Nr 30(1/8"+) drills. Got a buddy that does model airplanes or model trains? They're sure to have at least one on hand.
As an alternative, there are small (jewelers?) reamers that go down to a needle point. I have a set,
somewhere. They aren't as useful as the manufacturer claims. I'd rather use a small drill in a hand held chuck as a reamer. Just starting out a size or two smaller than desired. Getting it smooth by sliding the drill back and forth a few times. Then using the desired drill like a chucking reamer.
Just my two cents worth from working with model trains.
Bill Hudson