If I understand what you're suggesting, wouldn't this result in a lot more swinging since the gripper would be suspended from a single point directly above it?

You seem to. That is a good point I hadn't considered - the tension from the sides holds it in place. Maybe we need to use a telescoping arm (which doesn't seem practical, I'm mostly joking). It might still work, but I imagine if you don't grab it on the first try, you may have to go back to the ceiling. And hope there are no strong drafts in the room.

EDIT: maybe lower it on three lines instead of one?

That too is a solvable problem if you have the sensors for it. You can critically damp or choose a motion profile to have essentially no swing.