One thing we might draw parallels with would be "A Matter of Honor". The emergency signaling device there was essentially doing the job of a commbadge: calling for instant beam-up. It just did that silently.
With a commbadge, one can specify "three to beam up", and somehow O'Brien then manages to beam up the exact right three out of a crowd. Perhaps there's something more to the commbadge (which Riker also had in "AMoH") than meets the eye, then?
And perhaps the doodad from "AMoH" and the somewhat smaller doodad from ST:NEM are one and the same thing, lacking that mysterious something, and therefore only allowing one person without luggage to be sent over, hugging or no hugging?
In any case, regardless of the clumsiness or silliness of the setup, we shouldn't think in terms of "how to rescue Picard", because it's quite possible that rescue was never a serious consideration in that situation. Rescue was just a happy coincidence in the successful completion of the primary mission.