In the club's basic handgun class, I start students off shooting at a huge piece of butcher paper and just tell them to shoot at the middle of it. The idea is to familiarize people who have never fired a gun with the trigger feel, noise and recoil (or lack of it compared to what they might be expecting) before we even think about adding a target into the mix. If there is any flinching, trigger jerking or other basic problems I work through them at this time. Once the student is comfortable with safely making the gun go bang then we move on to a paper shopping bag for a target, then a sheet of notebook paper and finally down to a three-by-five inch card with a black aiming spot on it by the end of the session. Here, the sight picture really comes into play but we review all the basics at each step. It's all accuracy-related, eventually, because hitting what they are shooting at is the their purpose and the co-reason they are taking the class. The other reason is to learn how to do all this safely.

We do the air and .22 rifle instruction for hunter safety and 4H very much like you described. The emphasis, even as you described, is on the bullseye, here. And before we consider the students "instructed" we have them adjusting the sights as necessary to consistently hit the bullseye. Shooting groups is only a step on the way to the final goal. The actual purpose for shooting tight groups is to demonstrate that hitting the bullseye once is not just a fluke, we have developed the mechanical and mental ability to hammer it consistently. Not some spot near it, but "IT".