Here's something equally odd. The drill has two pulley arrangements for fast and slow speed. If you look at the first picture you see a largehand wheel on the side that raises and lowers the motor at an angle when it's turned. The handwheel has speeds on it ranging from 2850 / 880 at the slowest to 9350 / 2700 at the fastest. I'm sure the motor is not original but how would sliding the motor back and down change the speeds?
Maybe it had a variable speed pulley on there -- like a Roto-Cone or Lovejoy ? You can get them where only one side of the sheave moves, so moving the motor up/down on the incline would also shorten/lengthen the motor to pulley distance. Just a thought, but I have two of those variable drives and they are slick that way.
Hmm, interesting is right. Don't know if that would work then. I have one drive that uses a normal V-belt section, and the other which is considerably older -- circa 1920 -- uses a belt that is more flat than V. It's about an inch wide with much shallower angle on the sides, so kind of a hybrid. They're still made, Gates I think, or TB Woods. Whether something like that would work on your pulleys is hard to say without closer knowledge.
Plenty of drill presses move the whole head for adjustment. If its a gang or bench mount its usually the only way as the table is the base. Some are counterweighted or they will use a screw like this one. That's about the only thing on this machine I would actually consider normal.