Is it possible to add a Child Of modifier without changing position of the child? Currently, when I add a child-of modifier, the child jumps to the parent's head and rotates.



Why is that?

And more importantly, can I have blender not do that and keep the bone where and how it was, instead?

(I just want to "parent" a second armature to the first as if I had one armature where I parented with keep offset.)
Yes, in the constraint, click on the Set Inverse button. By default, the child will stick to its parent, but if you click Set Inverse it will keep its original position (like a Parent > Keep Offset).