Possibly, but the first really sharp blades were chipped out of stone in the Mesolithic at least. No one had to wait on the invention metallurgy, let alone steel. We find them from 18,000 BC. Some of the earliest bronze tools we find are razors.

You don't need a dedicated razor to shave (or to cut a long beard short). A sharpened knife or arrowhead will work.

You don't need to shave to get rid of hair. Plucking out the beard with clamshells used as tweezers was common in North America before the Europeans invaded.