How can the projection of uniform circular motion be SHM if UCM has no restoring force?

I'm confused about the relationship between simple harmonic motion (SHM) and uniform circular motion (UCM).

One of the conditions for SHM is that there must be a restoring force directed toward the equilibrium position, proportional to the displacement. However, we are also taught that SHM can be viewed as the projection of uniform circular motion onto a diameter.

My confusion is this: in UCM, the force acting on the particle is the centripetal force, not a restoring force in the SHM sense. So if there is no restoring force in the actual circular motion, how can its projection represent SHM, which fundamentally requires a restoring force?