The pneumatic tire is an awesome invention. They have the ability to cancel out imperfections in the road surface over a wide range of frequencies and amplitudes. Bicycles have traditionally used no other type of suspension.
To make a non-pneumatic tire you have to invent something equally as awesome. Normally you end up with something that produces a very rough ride.
A rough ride is the least of your problems. In a powered vehicle, the primary purpose of suspension is to keep you in contact with the road. If you're not in contact with the road, you're not in control; braking, steering and acceleration don't do anything if they can't react against the surface.
To make a non-pneumatic tire you have to invent something equally as awesome. Normally you end up with something that produces a very rough ride.