It's not even that realistic. In real life, you need to apply constant pressure downward to keep a machine gun firing level, but not constant movement.
Yeah, exactly. Pushing downwards slightly with a thumb stick / joystick might be a reasonable approximation. But, the whole thing you're trying to do is to prevent any movement.
But, I like your idea for the shifting aim point. If they ever try to do one for a mouse again, that would make more sense.