Why can a simply connected domain in 3D have a hole at the origin?
It's not hard to think of a path around the origin, and laying in a plane through the origin in 3D. How can we then "shrink" such a path to a point if the origin is not part of the domain?
Shouldn't we run into the same problem we have in the case of a 2D domain with a hole at the origin?
I'm clearly misunderstand something in the definition of a simply connected domain!
Simply connected refers to closed loops that can be contracted. In your case, such a loop can be contracted to a point in 3D no matter where you position the loop. Not so in 2D. The generalization you are thinking of is that of a closed surface: in your case you cannot contract just any closed surface in 3D. If a closed surface wraps around the hole at the origin it cannot be contracted to a point. But "simply connected" refers to curves and not surfaces. There are higher dimensional analogs of simply connected just the way you are thinking. See https://en.wikipedia.org/wiki/Homotopy and https://en.wikipedia.org/wiki/Homotopy_group