My third grandson is left handed, right eye dominant. He's also small for his age but very athletic. While it felt odd to him [at first] using a right hand finger on the trigger, he's done as well as the two older grandsons. With both hands on the rifle or handgun, and both eyes open, it wasn't a matter of arm strength to hold steady. He's 10 now, and likes shooting a .223 off the bench. But shooting from other positions he prefers to shoot .22lr and .22wmr as those rifles and handguns are more manageable for a kid his size.
Same here, mines 10 also and will shoot 17's and 22's all day but gets anxious and anticipates recoil/noise shooting center fire, even off a bench. I'm not pushing him, as long as he's shooting and enjoys it he'll come around to the bigger stuff when he's ready.