Agree...I've fixed many of these breaks in the shop..I wouldn't take it apart. The angle which that rung is broken you could just glue it back together with wood glue and put a spring clamp on it. You might have to put a bar clamp across the exterior to keep the joint from sliding apart. It will make a strong joint if done right.
Unless your replacing the spindle, there's no reason to dismantle it...
What that saying...." glue is stronger than wood itself"