But that's not what's happening. Placing items from chests (or the other way around) is a construction bot task, not a logistics bot task--they do it fine if there's a real roboport, but not if there's just a repair tower. And things get even stranger:
1) They can place from chests. They can pick up stuff and take to chests. But they can't upgrade items.
2) I laid out some power tiles--oops, I'm changing things around, those tiles are older, the parts exist but not in the logistics network. I grab enough of the missing parts and immediately trash them. I wasn't thinking enough and quickly walked out of roboport range before the bots had taken them. I'm out upgrading my defenses and bump into a repair tower--and the green beams show up. A bit of experimenting shows that while I'm bumped into a tower (note: I'm using squeak through, I might be getting closer than expected!) the beams can pull the parts from my trash slots. The repair tower is providing roboport coverage of range zero.