Anything I may be missing in my thinking here?
Yes! I went through this same thought process when I got my bolt gun... My thoughts:
You'd be releasing tension from your sling every time you run the bolt. Try keeping your sights on target using only your shoulder and firing hand... It sucks. I'd rather maintain sling tension, cheek weld, and shoulder pressure and just run that bolt with my firing hand... There is way less muzzle movement.