It depends upon the cowboy, didn’t it?

Cowboys normally grew up around horses and learned how to ride them in their pre-teen years. They knew horses (although nearly every outfit employed at least one wrangler who specialized in caring for the horses at the level just below a vet and a farrier), their livelihoods depending upon their ability to ride them, and they rarely went a day without mounting one. They were horsemen, although their abilities ranged from average to superior, depending upon their temperament, their judgment, and any disabilities that they may have experienced.

Marion Morrison (aka John Wayne) grew up in Iowa, moved to California when he was eight, and lived in the small towns (for the time) of Palmdale and Glendale. He apparently learned to ride in his teens and he demonstrated an aptitude for it. However he was never a cowboy, nor a ranch hand, and his livelihood (beyond acting) didn’t depend upon his riding like a professional. Feature films have always been fakes and the riding that Wayne is shown as doing was him mounting the animal, riding for the length of a take (or takes), and then dismounting. He didn’t ride for hours or days on end, he never went on cattle drives, and he certainly didn’t cross states on horseback like many cowboys and ranch hands did.

Would Wayne have made a good cowboy?

Probably not.

He never performed that work in his youth, nor when he was an adult and he probably didn’t ride every day when he wasn’t filming. Real cowboys spend hours or days in the saddle and there are no records of Wayne having done that. He might be able to ride well enough to keep a job, however, he was unlikely to have been a wrangler or ranch foreman.