In flexural design for concrete, is steel reinforcement required? Why?

Strictly speaking, no.

Without reinforcement, steel or otherwise, you are looking at gravity structures that mostly rely on compressive strength. Any modern application of concrete pretty much calls for reinforcement.

In different terms, once your concrete object (this is a solid concrete object) gets a dimension that is around three times more than the shortest, that’s about as far as you can go with unreinforced concrete. So, imagine a 1x1x3 object scaled as big as you want, that’s near your limit. No thin or narrow objects without your steel.