Having holes go oversized is not uncommon, and as you noticed, a misalignment can in fact cause them. Aluminum especially tends to drill oversized, especially when the hole is deep.
One way to avoid a small hole (in a small part) from going large is to drill it first perhaps 8 or 10 thousandths shy of the target, then remove it from the lathe. Switch the job to a drill press, and drill out the final hole by allowing the part to "float" a bit and find its own center. Another technique - if you've ever actually miked a small, wire-gauged drill, you'll perhaps have noticed that they mike slightly smaller than the hole they are supposed to drill. All drills actually cut a hole physically larger than the space that they occupy, otherwise, they'd simply stick in the hole! I hope this makes sense. The drills are ground to a specific dimension with the knowledge that the hole they will cut will be ever so slightly larger than the physical size of the drill. This effect, as a %, is more pronounced in the small wire gauge drills than a heftier drill bit.
What this means - If I have the mating part or pin, I'll often try sneaking up on the final hole by drilling it out one or two wire gauge sizes smaller, and physically checking with the mating part. For example, rather than drill straightaway with a 1/16" drill you might try a #53 first. If that works, you're done. If not, proceed to the 1/16" drill, and since you've pre-drilled, the subsequent hole should be closer to 0.0625" rather than oversized. HTH - I really can't see the need for a 1/16" reamer.
If you ultimately need to make a 1/16" d-bit or other homemade cutter, you might want to try a nice little piece of 1/16" music wire, and simply grind a 1/2 flute into the tip with a dremel cutoff wheel. It won't last, being spring-tempered carbon steel, but it should ream a hole or two in aluminum or brass.
One last option - take a 1/16" drill bit (they're cheap), chuck it in a drill press, and lightly apply a fine diamond or oil stone to the drill's flutes while it's spinning. You'll probably ruin the drill bit, but it's possible that a light stoning like that might cause it to drill 0.0005" or 0.001" less than it normally would. Obviously use it in a pre-drilled (undersized) hole, and check the outcome on scrap first. Good luck!
