A nonmetal would probably not be good for the frame of a bicycle because it would be too?

The question doesn't make much sense at the current state of engineering.

Decades ago, a nonmetal frame would have been likely to be either excessively clumsy and heavy, or too weak/flexy.

But with the introduction of carbon fiber, and even epoxy bonded wood(particularly bamboo), it's entirely possible to build perfectly functional nonmetal frames.

In fact, nonmetal in the shape of carbon fiber is the prime choice for high performance bicycles.