There are many ways of lifting an elevator.
If you look at some of the robot release videos you will see teams that lifted the elevator with a belt and with chain. Some lifted from the center and some teams lifted from both sides of the elevator.
We chose to use chain from both sides in a loop, to balance the lifting force side to side and to pull the elevator down, to reduce the chance of the elevator jamming. We also located our motor in the frame to lower the robot's center of gravity.
I understand that you are trying to keep things simple and quick to build.
So my suggestion follows a tow rope design. We used 2 AndyMark flanges with a small spacer and bought some thin high tensile rope and tied a loop. Inside the loop we tied our tow rope. I think it was a 1/4" Dacron (sailboat) rope. The sides of the flanges were enough to keep the rope from slipping off.
If you want to go with the belting, you can fold the belt over and sew it closed, then pass the thin rope thru and tie it to the flange. If the flanges are too small to contain the wraps, rivet a disk with a hole cutout to the inside of the flanges.
To keep things really simple, just use spectra line. Its real thin and super strong or just use the Dacron rope (line).
AndyMark sells hex and round bored flanges.
http://www.andymark.com/Hubs-s/215.htm
Rope
https://goodwinds.com/line-winders/b...ctra-line.html
http://www.westmarine.com/buy/new-en...ne--P012357679
http://www.westmarine.com/buy/fse-ro...e-49--12946836