It might be OK in the short term, depending on what your work is. I suspect it will be a poor speed/power match. And most drills are intermittent duty. Might not live very long.
Taig sells a step pulley set that uses a very thin belt that has worked great for me. I have the motor mounted on a hinge and lift it to change speeds. Very quick and easy. Why not use a DC motor and speed control or some type of universal motor with a router speed controller? Hint: of these two choices, DC is much quieter! Universal motor-think vacuum cleaner or router.
Harbor freight sells a 12 VDC pump that has a lot of torque and is on sale for about $25. I think it would be a good match and easy to adapt.
Or just pick up a junk treadmill and scavenge the motor and controller.
Ron Thompson On the Beautiful Florida Space Coast, right beside the Kennedy Space Center, USA