opalstar,
Your drive wheel is not big enough in diameter. A bit of tape won't last. And if whoever wrapped it didn't know which way it spun, it is likely to throw the tape off sooner.'
With the bigger wheel, it will remain in constant contact with the wheel and you won't have to hold the motor snug agaist it. Your motor is on springs to hold the tension. On any friction drive machine the drive wheel can develop flat spots from the motor being in the same position relative to the hand wheel if it stays that way for a long time.
One advantage of your spring mounted motor is that when you are not using the machine, you may wedge a spacer. ( clothes pin leg??) between the motor and the pillar to keep it from squishing into a flat spot on the drive wheel.