1. The workpiece of the required diameter is selected from the round timber. For large parts, this can be a piece of a bolt without thread, for small parts a piece of a regular nail.
2. We put the pulley on the shaft and at the boundary between the parts in the end we drill a hole to the desired depth.
3. We hammer a round billet into the hole and saw off the excess end with a hacksaw.
You realized that the veneers turned out to be semicircular, however, the mating can withstand significant loads. I was able to put even red-hot gears in this way, sawed a groove in a gear with a diamond-coated cable, designed for a bench saw, they are commercially available.