So, what happens when you strop on the smooth, nicked side? Have you tried gluing up the cuts with rubber cement?

Stropping is an acquired skill. It probably takes about a year before most new shavers are consistently improving an edge. Up to that point it probably does not make much difference.

You can improve the rough side of the leather by scraping it smooth with a cabinet card scraper, or large sharp kitchen knife. Lay the strop flat on a counter and take short light pass with the blade held 90 degrees to the strop. As you remove the uneven leather it will get smoother and you can take longer smoother strokes. If done properly, you will end up with a finish similar to a napped suede like finish.

Once your stropping improves, buy a better-quality strop and your edges will improve further.